from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul

from sympy.core.exprtools import factor_terms
from sympy.core.numbers import GoldenRatio
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.special.gamma_functions import gamma
from sympy.polys import PolynomialError, factor
from sympy.series.order import Order
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import together
from .gruntz import gruntz
from sympy.core.cache import cacheit


def limit(e, z, z0, direction=1):
    """Computes the limit of ``e(z)`` at the point ``z0``.

    Parameters
    ==========

    e : expression, the limit of which is to be taken

    z : symbol representing the variable in the limit.
        Other symbols are treated as constants. Multivariate limits
        are not supported.

    z0 : the value toward which ``z`` tends. Can be any expression,
        including ``oo`` and ``-oo``.

    direction : string, optional (default: "+")
        The limit is bi-directional if ``direction="+-"``, from the right
        (z->z0+) if ``direction="+"``, and from the left (z->z0-) if
        ``direction="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``direction``
        argument is determined from the direction of the infinity
        (i.e., ``direction="-"`` for ``oo``).

    Examples
    ========

    >>> from sympy import limit, sin, Symbol, oo
    >>> from sympy.abc import x
    >>> limit(sin(x)/x, x, 0)
    1
    >>> limit(1/x, x, 0) # default direction='+'
    oo
    >>> limit(1/x, x, 0, direction="-")
    -oo
    >>> limit(1/x, x, 0, direction='+-')
    Traceback (most recent call last):
        ...
    ValueError: The limit does not exist since left hand limit = -oo and right hand limit = oo

    >>> limit(1/x, x, oo)
    0

    Notes
    =====

    First we try some heuristics for easy and frequent cases like "x", "1/x",
    "x**2" and similar, so that it's fast. For all other cases, we use the
    Gruntz algorithm (see the gruntz() function).

    See Also
    ========

     limit_seq : returns the limit of a sequence.
    """

    if direction == 0:
        llim = Limit[z:z0:-1](e).doit(deep=False)
        rlim = Limit[z:z0:1](e).doit(deep=False)
        if llim == rlim:
            return rlim
        else:
            # TODO: choose a better error?
            raise ValueError("The limit does not exist since "
                    "left hand limit = %s and right hand limit = %s"
                    % (llim, rlim))
    else:
        return Limit[z:z0:direction](e).doit(deep=False)


def heuristics(e, z, z0, direction):
    """Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    """

    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1 / z), z, S.Zero, 1 if z0 is S.Infinity else -1)
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, direction)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul):  # try together
                        m = together(m)
                    if not isinstance(m, Mul):  # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, direction)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    from sympy import simplify
                    e3 = simplify(Mul(*e2))
                    l = limit(e3, z, z0, direction)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, direction)
    return rv


class Limit(Expr):
    """Represents an unevaluated limit.

    Examples
    ========

    >>> from sympy import Limit, sin, Symbol
    >>> from sympy.abc import x
    >>> Limit(sin(x)/x, x, 0)
    Limit(sin(x)/x, x, 0)
    >>> Limit(1/x, x, 0, direction="-")
    Limit(1/x, x, 0, direction='-')

    """

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        
        limit = limits[0]
        if len(limit) == 3: 
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
            
        e = sympify(e)
        z = sympify(z)
        
        assert z.is_symbol and not z.is_given
        
        z0 = sympify(z0)

        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero

        assert direction in (1, 0, -1), "direction must be one of 1, 0, -1, not %s" % direction

        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = (e, Tuple(z, z0, direction))
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms

    def doit(self, **hints):
        """Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        """
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial

        e, (z, z0, direction) = self.args

        if z0 is S.ComplexInfinity:
            raise NotImplementedError("Limits at complex "
                                    "infinity are not implemented")

        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)

        if e == z:
            return z0

        if not e.has(z):
            return e

        # gruntz fails on factorials but works with the gamma function
        # If no factorial term is present, e should remain unchanged.
        # factorial is defined to be zero for negative inputs (which
        # differs from gamma) so only rewrite for positive z0.
        if z0.is_extended_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)

        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: (z in w.free_symbols and
                                any(a.is_polynomial(z) or
                                    any(z in m.free_symbols and m.is_polynomial(z)
                                        for m in Mul.make_args(a))
                                    for a in Add.make_args(w)))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=True)
                    if z0 is S.NegativeInfinity:
                        inve = e.subs(z, -1 / u)
                    else:
                        inve = e.subs(z, 1 / u)
                    try:
                        r = limit(inve.as_leading_term(u), u, S.Zero, 1)
                        if isinstance(r, Limit):
                            return self
                        else:
                            return r
                    except ValueError:
                        pass

        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1])

        try:
            if direction == 0:
                r = gruntz(e, z, z0, 1)
                _r = gruntz(e, z, z0, -1)
                if r != _r:
                    raise PoleError()
            else:
                r = gruntz(e, z, z0, direction)
                if r is S.NaN:
                    raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, direction)
            if r is None:
                return self
        except NotImplementedError:
            return self
        return r

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex        
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set
    
    @property
    def shape(self): 
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return "lim[%s>%s](%s)" % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return "lim[%s<%s](%s)" % tuple(map(p._print, (z, z0, e)))
        else:
            return "lim[%s:%s](%s)" % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args

        tex = r"\lim\limits_{%s \to " % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += r"%s}" % p._print(z0)
        else:
            tex += r"%s^%s}" % (p._print(z0), p._print('+' if dir > 0 else '-'))

        if isinstance(e, Add):
#         if isinstance(e, AssocOp):
            return r"%s\left(%s\right)" % (tex, p._print(e))
        else:
            return r"%s %s" % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm

        E = p._print(e)
        if precedence(e) <= PRECEDENCE["Mul"]:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')

        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))

        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ""
        else:
            if p._use_unicode:
                dir = u'\N{SUPERSCRIPT PLUS SIGN}' if dir == 1 else u'\N{SUPERSCRIPT MINUS}'

        LimArg = prettyForm(*LimArg.right(p._print(dir)))

        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)

        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        
        if expr.is_symbol:
            if expr == x:
                return x0
            
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                
                return Limit[x:x0:dir](expr) + const
            
        return self

    @cacheit
    def _has(self, pattern):
        """Helper for .has()"""        
        from sympy.core.assumptions import BasicMeta
        from sympy.core.function import UndefinedFunction
        if isinstance(pattern, (BasicMeta, UndefinedFunction)):
            return Expr._has(self, pattern)
        
        from sympy.tensor.indexed import Indexed, Sliced
        if not isinstance(pattern, (Symbol, Indexed, Sliced)):
            return Expr._has(self, pattern)

        expr = self.expr
        limits = []
        
        for limit in self.limits:
            x, *_ = limit
            if x == pattern:
                return False
            
            limits.append(limit)
                    
        boolean = expr._has(pattern)

        return boolean or any(arg._has(pattern) for arg in limits)

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        
        return Expr._subs(self, old, new, **hints)
    
    def _eval_is_super_real(self):
        return self.expr.is_super_real
f84b234f385d58aae1515366e916270a2b251a4a
<mask token>


class Limit(Expr):
    <mask token>

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        limit = limits[0]
        if len(limit) == 3:
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
        e = sympify(e)
        z = sympify(z)
        assert z.is_symbol and not z.is_given
        z0 = sympify(z0)
        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero
        assert direction in (1, 0, -1
            ), 'direction must be one of 1, 0, -1, not %s' % direction
        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = e, Tuple(z, z0, direction)
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms
    <mask token>

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set

    @property
    def shape(self):
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return 'lim[%s>%s](%s)' % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return 'lim[%s<%s](%s)' % tuple(map(p._print, (z, z0, e)))
        else:
            return 'lim[%s:%s](%s)' % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args
        tex = '\\lim\\limits_{%s \\to ' % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += '%s}' % p._print(z0)
        else:
            tex += '%s^%s}' % (p._print(z0), p._print('+' if dir > 0 else '-'))
        if isinstance(e, Add):
            return '%s\\left(%s\\right)' % (tex, p._print(e))
        else:
            return '%s %s' % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm
        E = p._print(e)
        if precedence(e) <= PRECEDENCE['Mul']:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')
        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'─→'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ''
        elif p._use_unicode:
            dir = u'⁺' if dir == 1 else u'⁻'
        LimArg = prettyForm(*LimArg.right(p._print(dir)))
        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        if expr.is_symbol:
            if expr == x:
                return x0
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                return Limit[x:x0:dir](expr) + const
        return self
    <mask token>

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        return Expr._subs(self, old, new, **hints)
    <mask token>

<mask token>


class Limit(Expr):
    <mask token>

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        limit = limits[0]
        if len(limit) == 3:
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
        e = sympify(e)
        z = sympify(z)
        assert z.is_symbol and not z.is_given
        z0 = sympify(z0)
        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero
        assert direction in (1, 0, -1
            ), 'direction must be one of 1, 0, -1, not %s' % direction
        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = e, Tuple(z, z0, direction)
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms

    def doit(self, **hints):
        """Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        """
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial
        e, (z, z0, direction) = self.args
        if z0 is S.ComplexInfinity:
            raise NotImplementedError(
                'Limits at complex infinity are not implemented')
        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)
        if e == z:
            return z0
        if not e.has(z):
            return e
        if z0.is_extended_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)
        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: z in w.free_symbols and any(a.is_polynomial(
                    z) or any(z in m.free_symbols and m.is_polynomial(z) for
                    m in Mul.make_args(a)) for a in Add.make_args(w))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=True)
                    if z0 is S.NegativeInfinity:
                        inve = e.subs(z, -1 / u)
                    else:
                        inve = e.subs(z, 1 / u)
                    try:
                        r = limit(inve.as_leading_term(u), u, S.Zero, 1)
                        if isinstance(r, Limit):
                            return self
                        else:
                            return r
                    except ValueError:
                        pass
        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1])
        try:
            if direction == 0:
                r = gruntz(e, z, z0, 1)
                _r = gruntz(e, z, z0, -1)
                if r != _r:
                    raise PoleError()
            else:
                r = gruntz(e, z, z0, direction)
                if r is S.NaN:
                    raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, direction)
            if r is None:
                return self
        except NotImplementedError:
            return self
        return r

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set

    @property
    def shape(self):
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return 'lim[%s>%s](%s)' % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return 'lim[%s<%s](%s)' % tuple(map(p._print, (z, z0, e)))
        else:
            return 'lim[%s:%s](%s)' % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args
        tex = '\\lim\\limits_{%s \\to ' % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += '%s}' % p._print(z0)
        else:
            tex += '%s^%s}' % (p._print(z0), p._print('+' if dir > 0 else '-'))
        if isinstance(e, Add):
            return '%s\\left(%s\\right)' % (tex, p._print(e))
        else:
            return '%s %s' % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm
        E = p._print(e)
        if precedence(e) <= PRECEDENCE['Mul']:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')
        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'─→'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ''
        elif p._use_unicode:
            dir = u'⁺' if dir == 1 else u'⁻'
        LimArg = prettyForm(*LimArg.right(p._print(dir)))
        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        if expr.is_symbol:
            if expr == x:
                return x0
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                return Limit[x:x0:dir](expr) + const
        return self
    <mask token>

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        return Expr._subs(self, old, new, **hints)
    <mask token>

<mask token>


class Limit(Expr):
    """Represents an unevaluated limit.

    Examples
    ========

    >>> from sympy import Limit, sin, Symbol
    >>> from sympy.abc import x
    >>> Limit(sin(x)/x, x, 0)
    Limit(sin(x)/x, x, 0)
    >>> Limit(1/x, x, 0, direction="-")
    Limit(1/x, x, 0, direction='-')

    """

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        limit = limits[0]
        if len(limit) == 3:
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
        e = sympify(e)
        z = sympify(z)
        assert z.is_symbol and not z.is_given
        z0 = sympify(z0)
        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero
        assert direction in (1, 0, -1
            ), 'direction must be one of 1, 0, -1, not %s' % direction
        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = e, Tuple(z, z0, direction)
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms

    def doit(self, **hints):
        """Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        """
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial
        e, (z, z0, direction) = self.args
        if z0 is S.ComplexInfinity:
            raise NotImplementedError(
                'Limits at complex infinity are not implemented')
        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)
        if e == z:
            return z0
        if not e.has(z):
            return e
        if z0.is_extended_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)
        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: z in w.free_symbols and any(a.is_polynomial(
                    z) or any(z in m.free_symbols and m.is_polynomial(z) for
                    m in Mul.make_args(a)) for a in Add.make_args(w))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=True)
                    if z0 is S.NegativeInfinity:
                        inve = e.subs(z, -1 / u)
                    else:
                        inve = e.subs(z, 1 / u)
                    try:
                        r = limit(inve.as_leading_term(u), u, S.Zero, 1)
                        if isinstance(r, Limit):
                            return self
                        else:
                            return r
                    except ValueError:
                        pass
        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1])
        try:
            if direction == 0:
                r = gruntz(e, z, z0, 1)
                _r = gruntz(e, z, z0, -1)
                if r != _r:
                    raise PoleError()
            else:
                r = gruntz(e, z, z0, direction)
                if r is S.NaN:
                    raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, direction)
            if r is None:
                return self
        except NotImplementedError:
            return self
        return r

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set

    @property
    def shape(self):
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return 'lim[%s>%s](%s)' % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return 'lim[%s<%s](%s)' % tuple(map(p._print, (z, z0, e)))
        else:
            return 'lim[%s:%s](%s)' % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args
        tex = '\\lim\\limits_{%s \\to ' % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += '%s}' % p._print(z0)
        else:
            tex += '%s^%s}' % (p._print(z0), p._print('+' if dir > 0 else '-'))
        if isinstance(e, Add):
            return '%s\\left(%s\\right)' % (tex, p._print(e))
        else:
            return '%s %s' % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm
        E = p._print(e)
        if precedence(e) <= PRECEDENCE['Mul']:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')
        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'─→'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ''
        elif p._use_unicode:
            dir = u'⁺' if dir == 1 else u'⁻'
        LimArg = prettyForm(*LimArg.right(p._print(dir)))
        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        if expr.is_symbol:
            if expr == x:
                return x0
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                return Limit[x:x0:dir](expr) + const
        return self

    @cacheit
    def _has(self, pattern):
        """Helper for .has()"""
        from sympy.core.assumptions import BasicMeta
        from sympy.core.function import UndefinedFunction
        if isinstance(pattern, (BasicMeta, UndefinedFunction)):
            return Expr._has(self, pattern)
        from sympy.tensor.indexed import Indexed, Sliced
        if not isinstance(pattern, (Symbol, Indexed, Sliced)):
            return Expr._has(self, pattern)
        expr = self.expr
        limits = []
        for limit in self.limits:
            x, *_ = limit
            if x == pattern:
                return False
            limits.append(limit)
        boolean = expr._has(pattern)
        return boolean or any(arg._has(pattern) for arg in limits)

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        return Expr._subs(self, old, new, **hints)

    def _eval_is_super_real(self):
        return self.expr.is_super_real

<mask token>


def limit(e, z, z0, direction=1):
    """Computes the limit of ``e(z)`` at the point ``z0``.

    Parameters
    ==========

    e : expression, the limit of which is to be taken

    z : symbol representing the variable in the limit.
        Other symbols are treated as constants. Multivariate limits
        are not supported.

    z0 : the value toward which ``z`` tends. Can be any expression,
        including ``oo`` and ``-oo``.

    direction : string, optional (default: "+")
        The limit is bi-directional if ``direction="+-"``, from the right
        (z->z0+) if ``direction="+"``, and from the left (z->z0-) if
        ``direction="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``direction``
        argument is determined from the direction of the infinity
        (i.e., ``direction="-"`` for ``oo``).

    Examples
    ========

    >>> from sympy import limit, sin, Symbol, oo
    >>> from sympy.abc import x
    >>> limit(sin(x)/x, x, 0)
    1
    >>> limit(1/x, x, 0) # default direction='+'
    oo
    >>> limit(1/x, x, 0, direction="-")
    -oo
    >>> limit(1/x, x, 0, direction='+-')
    Traceback (most recent call last):
        ...
    ValueError: The limit does not exist since left hand limit = -oo and right hand limit = oo

    >>> limit(1/x, x, oo)
    0

    Notes
    =====

    First we try some heuristics for easy and frequent cases like "x", "1/x",
    "x**2" and similar, so that it's fast. For all other cases, we use the
    Gruntz algorithm (see the gruntz() function).

    See Also
    ========

     limit_seq : returns the limit of a sequence.
    """
    if direction == 0:
        llim = Limit[z:z0:-1](e).doit(deep=False)
        rlim = Limit[z:z0:1](e).doit(deep=False)
        if llim == rlim:
            return rlim
        else:
            raise ValueError(
                'The limit does not exist since left hand limit = %s and right hand limit = %s'
                 % (llim, rlim))
    else:
        return Limit[z:z0:direction](e).doit(deep=False)


def heuristics(e, z, z0, direction):
    """Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    """
    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1 / z), z, S.Zero, 1 if z0 is S.Infinity else -1)
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, direction)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul):
                        m = together(m)
                    if not isinstance(m, Mul):
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, direction)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for
                rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])
                if len(e2) > 0:
                    from sympy import simplify
                    e3 = simplify(Mul(*e2))
                    l = limit(e3, z, z0, direction)
                    rv = l * Mul(*r2)
            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, direction)
    return rv


class Limit(Expr):
    """Represents an unevaluated limit.

    Examples
    ========

    >>> from sympy import Limit, sin, Symbol
    >>> from sympy.abc import x
    >>> Limit(sin(x)/x, x, 0)
    Limit(sin(x)/x, x, 0)
    >>> Limit(1/x, x, 0, direction="-")
    Limit(1/x, x, 0, direction='-')

    """

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        limit = limits[0]
        if len(limit) == 3:
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
        e = sympify(e)
        z = sympify(z)
        assert z.is_symbol and not z.is_given
        z0 = sympify(z0)
        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero
        assert direction in (1, 0, -1
            ), 'direction must be one of 1, 0, -1, not %s' % direction
        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = e, Tuple(z, z0, direction)
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms

    def doit(self, **hints):
        """Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        """
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial
        e, (z, z0, direction) = self.args
        if z0 is S.ComplexInfinity:
            raise NotImplementedError(
                'Limits at complex infinity are not implemented')
        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)
        if e == z:
            return z0
        if not e.has(z):
            return e
        if z0.is_extended_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)
        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: z in w.free_symbols and any(a.is_polynomial(
                    z) or any(z in m.free_symbols and m.is_polynomial(z) for
                    m in Mul.make_args(a)) for a in Add.make_args(w))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=True)
                    if z0 is S.NegativeInfinity:
                        inve = e.subs(z, -1 / u)
                    else:
                        inve = e.subs(z, 1 / u)
                    try:
                        r = limit(inve.as_leading_term(u), u, S.Zero, 1)
                        if isinstance(r, Limit):
                            return self
                        else:
                            return r
                    except ValueError:
                        pass
        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1])
        try:
            if direction == 0:
                r = gruntz(e, z, z0, 1)
                _r = gruntz(e, z, z0, -1)
                if r != _r:
                    raise PoleError()
            else:
                r = gruntz(e, z, z0, direction)
                if r is S.NaN:
                    raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, direction)
            if r is None:
                return self
        except NotImplementedError:
            return self
        return r

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set

    @property
    def shape(self):
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return 'lim[%s>%s](%s)' % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return 'lim[%s<%s](%s)' % tuple(map(p._print, (z, z0, e)))
        else:
            return 'lim[%s:%s](%s)' % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args
        tex = '\\lim\\limits_{%s \\to ' % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += '%s}' % p._print(z0)
        else:
            tex += '%s^%s}' % (p._print(z0), p._print('+' if dir > 0 else '-'))
        if isinstance(e, Add):
            return '%s\\left(%s\\right)' % (tex, p._print(e))
        else:
            return '%s %s' % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm
        E = p._print(e)
        if precedence(e) <= PRECEDENCE['Mul']:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')
        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'─→'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ''
        elif p._use_unicode:
            dir = u'⁺' if dir == 1 else u'⁻'
        LimArg = prettyForm(*LimArg.right(p._print(dir)))
        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        if expr.is_symbol:
            if expr == x:
                return x0
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                return Limit[x:x0:dir](expr) + const
        return self

    @cacheit
    def _has(self, pattern):
        """Helper for .has()"""
        from sympy.core.assumptions import BasicMeta
        from sympy.core.function import UndefinedFunction
        if isinstance(pattern, (BasicMeta, UndefinedFunction)):
            return Expr._has(self, pattern)
        from sympy.tensor.indexed import Indexed, Sliced
        if not isinstance(pattern, (Symbol, Indexed, Sliced)):
            return Expr._has(self, pattern)
        expr = self.expr
        limits = []
        for limit in self.limits:
            x, *_ = limit
            if x == pattern:
                return False
            limits.append(limit)
        boolean = expr._has(pattern)
        return boolean or any(arg._has(pattern) for arg in limits)

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        return Expr._subs(self, old, new, **hints)

    def _eval_is_super_real(self):
        return self.expr.is_super_real

from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul

from sympy.core.exprtools import factor_terms
from sympy.core.numbers import GoldenRatio
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.special.gamma_functions import gamma
from sympy.polys import PolynomialError, factor
from sympy.series.order import Order
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import together
from .gruntz import gruntz
from sympy.core.cache import cacheit


def limit(e, z, z0, direction=1):
    """Computes the limit of ``e(z)`` at the point ``z0``.

    Parameters
    ==========

    e : expression, the limit of which is to be taken

    z : symbol representing the variable in the limit.
        Other symbols are treated as constants. Multivariate limits
        are not supported.

    z0 : the value toward which ``z`` tends. Can be any expression,
        including ``oo`` and ``-oo``.

    direction : string, optional (default: "+")
        The limit is bi-directional if ``direction="+-"``, from the right
        (z->z0+) if ``direction="+"``, and from the left (z->z0-) if
        ``direction="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``direction``
        argument is determined from the direction of the infinity
        (i.e., ``direction="-"`` for ``oo``).

    Examples
    ========

    >>> from sympy import limit, sin, Symbol, oo
    >>> from sympy.abc import x
    >>> limit(sin(x)/x, x, 0)
    1
    >>> limit(1/x, x, 0) # default direction='+'
    oo
    >>> limit(1/x, x, 0, direction="-")
    -oo
    >>> limit(1/x, x, 0, direction='+-')
    Traceback (most recent call last):
        ...
    ValueError: The limit does not exist since left hand limit = -oo and right hand limit = oo

    >>> limit(1/x, x, oo)
    0

    Notes
    =====

    First we try some heuristics for easy and frequent cases like "x", "1/x",
    "x**2" and similar, so that it's fast. For all other cases, we use the
    Gruntz algorithm (see the gruntz() function).

    See Also
    ========

     limit_seq : returns the limit of a sequence.
    """

    if direction == 0:
        llim = Limit[z:z0:-1](e).doit(deep=False)
        rlim = Limit[z:z0:1](e).doit(deep=False)
        if llim == rlim:
            return rlim
        else:
            # TODO: choose a better error?
            raise ValueError("The limit does not exist since "
                    "left hand limit = %s and right hand limit = %s"
                    % (llim, rlim))
    else:
        return Limit[z:z0:direction](e).doit(deep=False)


def heuristics(e, z, z0, direction):
    """Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    """

    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1 / z), z, S.Zero, 1 if z0 is S.Infinity else -1)
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, direction)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul):  # try together
                        m = together(m)
                    if not isinstance(m, Mul):  # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, direction)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    from sympy import simplify
                    e3 = simplify(Mul(*e2))
                    l = limit(e3, z, z0, direction)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, direction)
    return rv


class Limit(Expr):
    """Represents an unevaluated limit.

    Examples
    ========

    >>> from sympy import Limit, sin, Symbol
    >>> from sympy.abc import x
    >>> Limit(sin(x)/x, x, 0)
    Limit(sin(x)/x, x, 0)
    >>> Limit(1/x, x, 0, direction="-")
    Limit(1/x, x, 0, direction='-')

    """

    def __new__(cls, e, *limits):
        assert len(limits) == 1
        
        limit = limits[0]
        if len(limit) == 3: 
            z, z0, direction = limit
        else:
            z, z0 = limit
            direction = S.Zero
            
        e = sympify(e)
        z = sympify(z)
        
        assert z.is_symbol and not z.is_given
        
        z0 = sympify(z0)

        if z0 is S.Infinity:
            direction = S.NegativeOne
        elif z0 is S.NegativeInfinity:
            direction = S.One
        elif z.shape:
            direction = S.Zero

        assert direction in (1, 0, -1), "direction must be one of 1, 0, -1, not %s" % direction

        obj = Expr.__new__(cls)
        from sympy import Tuple
        obj._args = (e, Tuple(z, z0, direction))
        return obj

    @property
    def free_symbols(self):
        e, (z, z0, direction) = self.args
        isyms = e.free_symbols
        isyms.difference_update(z.free_symbols)
        isyms.update(z0.free_symbols)
        return isyms

    def doit(self, **hints):
        """Evaluates the limit.

        Parameters
        ==========

        deep : bool, optional (default: True)
            Invoke the ``doit`` method of the expressions involved before
            taking the limit.

        hints : optional keyword arguments
            To be passed to ``doit`` methods; only used if deep is True.
        """
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial

        e, (z, z0, direction) = self.args

        if z0 is S.ComplexInfinity:
            raise NotImplementedError("Limits at complex "
                                    "infinity are not implemented")

        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)

        if e == z:
            return z0

        if not e.has(z):
            return e

        # gruntz fails on factorials but works with the gamma function
        # If no factorial term is present, e should remain unchanged.
        # factorial is defined to be zero for negative inputs (which
        # differs from gamma) so only rewrite for positive z0.
        if z0.is_extended_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)

        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: (z in w.free_symbols and
                                any(a.is_polynomial(z) or
                                    any(z in m.free_symbols and m.is_polynomial(z)
                                        for m in Mul.make_args(a))
                                    for a in Add.make_args(w)))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=True)
                    if z0 is S.NegativeInfinity:
                        inve = e.subs(z, -1 / u)
                    else:
                        inve = e.subs(z, 1 / u)
                    try:
                        r = limit(inve.as_leading_term(u), u, S.Zero, 1)
                        if isinstance(r, Limit):
                            return self
                        else:
                            return r
                    except ValueError:
                        pass

        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1])

        try:
            if direction == 0:
                r = gruntz(e, z, z0, 1)
                _r = gruntz(e, z, z0, -1)
                if r != _r:
                    raise PoleError()
            else:
                r = gruntz(e, z, z0, direction)
                if r is S.NaN:
                    raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, direction)
            if r is None:
                return self
        except NotImplementedError:
            return self
        return r

    @property
    def dtype(self):
        expr = self.expr
        if expr.is_set:
            return expr.dtype
        
        from sympy import dtype
        if not expr.is_real:
            return dtype.complex        
        return dtype.real

    @property
    def is_set(self):
        return self.expr.is_set
    
    @property
    def shape(self): 
        return self.args[0].shape

    @property
    def expr(self):
        return self.args[0]

    @property
    def limits(self):
        return self.args[1:]

    @property
    def variable(self):
        return self.args[1][0]

    def _sympystr(self, p):
        e, (z, z0, direction) = self.args
        if direction == 1:
            return "lim[%s>%s](%s)" % tuple(map(p._print, (z, z0, e)))
        elif direction == -1:
            return "lim[%s<%s](%s)" % tuple(map(p._print, (z, z0, e)))
        else:
            return "lim[%s:%s](%s)" % tuple(map(p._print, (z, z0, e)))

    def _latex(self, p):
        e, (z, z0, dir) = self.args

        tex = r"\lim\limits_{%s \to " % p._print(z)
        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            tex += r"%s}" % p._print(z0)
        else:
            tex += r"%s^%s}" % (p._print(z0), p._print('+' if dir > 0 else '-'))

        if isinstance(e, Add):
#         if isinstance(e, AssocOp):
            return r"%s\left(%s\right)" % (tex, p._print(e))
        else:
            return r"%s %s" % (tex, p._print(e))

    def _pretty(self, p):
        e, (z, z0, dir) = self.args
        from sympy.printing.precedence import precedence, PRECEDENCE
        from sympy.printing.pretty.stringpict import prettyForm

        E = p._print(e)
        if precedence(e) <= PRECEDENCE["Mul"]:
            E = prettyForm(*E.parens('(', ')'))
        Lim = prettyForm('lim')

        LimArg = p._print(z)
        if p._use_unicode:
            LimArg = prettyForm(*LimArg.right(u'\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
        else:
            LimArg = prettyForm(*LimArg.right('->'))
        LimArg = prettyForm(*LimArg.right(p._print(z0)))

        if dir == 0 or z0 in (S.Infinity, S.NegativeInfinity):
            dir = ""
        else:
            if p._use_unicode:
                dir = u'\N{SUPERSCRIPT PLUS SIGN}' if dir == 1 else u'\N{SUPERSCRIPT MINUS}'

        LimArg = prettyForm(*LimArg.right(p._print(dir)))

        Lim = prettyForm(*Lim.below(LimArg))
        Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)

        return Lim

    def simplify(self, **kwargs):
        expr, (x, x0, dir) = self.args
        if not expr._has(x):
            return expr
        
        if expr.is_symbol:
            if expr == x:
                return x0
            
        if expr.is_Add:
            const = []
            args = []
            for arg in expr.args:
                if arg._has(x):
                    args.append(arg)
                else:
                    const.append(arg)
            if const:
                expr = Add(*args)
                const = Add(*const)
                if expr == x:
                    return x0 + const
                
                return Limit[x:x0:dir](expr) + const
            
        return self

    @cacheit
    def _has(self, pattern):
        """Helper for .has()"""        
        from sympy.core.assumptions import BasicMeta
        from sympy.core.function import UndefinedFunction
        if isinstance(pattern, (BasicMeta, UndefinedFunction)):
            return Expr._has(self, pattern)
        
        from sympy.tensor.indexed import Indexed, Sliced
        if not isinstance(pattern, (Symbol, Indexed, Sliced)):
            return Expr._has(self, pattern)

        expr = self.expr
        limits = []
        
        for limit in self.limits:
            x, *_ = limit
            if x == pattern:
                return False
            
            limits.append(limit)
                    
        boolean = expr._has(pattern)

        return boolean or any(arg._has(pattern) for arg in limits)

    def _subs(self, old, new, **hints):
        if old == self.variable:
            return self
        
        return Expr._subs(self, old, new, **hints)
    
    def _eval_is_super_real(self):
        return self.expr.is_super_real