from __future__ import print_function, division
from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul
from sympy.core.compatibility import string_types
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
[docs]def limit(e, z, z0, dir="+"):
"""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``.
dir : string, optional (default: "+")
The limit is bi-directional if ``dir="+-"``, from the right
(z->z0+) if ``dir="+"``, and from the left (z->z0-) if
``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir``
argument is determined from the direction of the infinity
(i.e., ``dir="-"`` 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 dir='+'
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, 0, dir='+-')
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 dir == "+-":
llim = Limit(e, z, z0, dir="-").doit(deep=False)
rlim = Limit(e, z, z0, dir="+").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(e, z, z0, dir).doit(deep=False)
def heuristics(e, z, z0, dir):
"""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, "+" if z0 is S.Infinity else "-")
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, dir)
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, dir)
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:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
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, dir)
return rv
[docs]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, dir="-")
Limit(1/x, x, 0, dir='-')
"""
def __new__(cls, e, z, z0, dir="+"):
e = sympify(e)
z = sympify(z)
z0 = sympify(z0)
if z0 is S.Infinity:
dir = "-"
elif z0 is S.NegativeInfinity:
dir = "+"
if isinstance(dir, string_types):
dir = Symbol(dir)
elif not isinstance(dir, Symbol):
raise TypeError("direction must be of type basestring or "
"Symbol, not %s" % type(dir))
if str(dir) not in ('+', '-', '+-'):
raise ValueError("direction must be one of '+', '-' "
"or '+-', not %s" % dir)
obj = Expr.__new__(cls)
obj._args = (e, z, z0, dir)
return obj
@property
def free_symbols(self):
e = self.args[0]
isyms = e.free_symbols
isyms.difference_update(self.args[1].free_symbols)
isyms.update(self.args[2].free_symbols)
return isyms
[docs] 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, dir = 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_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, "+")
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:
r = gruntz(e, z, z0, dir)
if r is S.NaN:
raise PoleError()
except (PoleError, ValueError):
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
