/

# Source code for sympy.series.limits

from sympy.core import S, Add, sympify, Expr, PoleError, Mul, oo, C
from gruntz import gruntz
from sympy.functions import sign, tan, cot

[docs]def limit(e, z, z0, dir="+"):
"""
Compute the limit of e(z) at the point z0.

z0 can be any expression, including oo and -oo.

For dir="+" (default) it calculates the limit from the right
(z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0
(oo or -oo), the dir argument doesn't matter.

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, dir="+")
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, oo)
0

Strategy:

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).
"""
from sympy import Wild, log

e = sympify(e)
z = sympify(z)
z0 = sympify(z0)

if e == z:
return z0

if e.is_Rational:
return e

if not e.has(z):
return e

if e.func is tan:
# discontinuity at odd multiples of pi/2; 0 at even
disc = S.Pi/2
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if i.is_even:
return S.Zero
elif i.is_odd:
if dir == '+':
return S.NegativeInfinity
else:
return S.Infinity

if e.func is cot:
# discontinuity at multiples of pi; 0 at odd pi/2 multiples
disc = S.Pi
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if dir == '-':
return S.NegativeInfinity
else:
return S.Infinity
elif (2*i).is_integer:
return S.Zero

if e.is_Pow:
b, ex = e.args
c = None # records sign of b if b is +/-z or has a bounded value
if b.is_Mul:
c, b = b.as_two_terms()
if c is S.NegativeOne and b == z:
c = '-'
elif b == z:
c = '+'

if ex.is_number:
if c is None:
base = b.subs(z, z0)
if base.is_bounded and (ex.is_bounded or base is not S.One):
return base**ex
else:
if z0 == 0 and ex < 0:
if dir != c:
# integer
if ex.is_even:
return S.Infinity
elif ex.is_odd:
return S.NegativeInfinity
# rational
elif ex.is_Rational:
return (S.NegativeOne**ex)*S.Infinity
else:
return S.ComplexInfinity
return S.Infinity
return z0**ex

if e.is_Mul or not z0 and e.is_Pow and b.func is log:
if e.is_Mul:
# weed out the z-independent terms
i, d = e.as_independent(z)
if i is not S.One and i.is_bounded:
return i*limit(d, z, z0, dir)
else:
i, d = S.One, e
if not z0:
# look for log(z)**q or z**p*log(z)**q
p, q = Wild("p"), Wild("q")
r = d.match(z**p * log(z)**q)
if r:
p, q = [r.get(w, w) for w in [p, q]]
if q and q.is_number and p.is_number:
if q > 0:
if p > 0:
return S.Zero
else:
return -oo*i
else:
if p >= 0:
return S.Zero
else:
return -oo*i

if e.is_polynomial() and not z0.is_unbounded:
return Add(*[limit(term, z, z0, dir) for term in e.args])

# this is a case like limit(x*y+x*z, z, 2) == x*y+2*x
# but we need to make sure, that the general gruntz() algorithm is
# executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0"
unbounded = []; unbounded_result=[]
finite = []; unknown = []
ok = True
for term in e.args:
if not term.has(z) and not term.is_unbounded:
finite.append(term)
continue
result = term.subs(z, z0)
bounded = result.is_bounded
if bounded is False or result is S.NaN:
if unknown:
ok = False
break
unbounded.append(term)
if result != S.NaN:
# take result from direction given
result = limit(term, z, z0, dir)
unbounded_result.append(result)
elif bounded:
finite.append(result)
else:
if unbounded:
ok = False
break
unknown.append(result)
if not ok:
# we won't be able to resolve this with unbounded
# terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -> oo:
# since the Sum is unevaluated it's boundedness is
# unknown and the log(n) is oo so you get Sum - oo
# which is unsatisfactory.
raise NotImplementedError('unknown boundedness for %s' %
(unknown or result))
if unbounded:
if inf_limit is not S.NaN:
return inf_limit + u
if finite:
else:

if e.is_Order:
args = e.args
return C.Order(limit(args[0], z, z0), *args[1:])

try:
r = gruntz(e, z, z0, dir)
if r is S.NaN:
raise PoleError()
except PoleError:
r = heuristics(e, z, z0, dir)
return r

def heuristics(e, z, z0, dir):
if z0 == oo:
return limit(e.subs(z, 1/z), z, sympify(0), "+")
elif e.is_Mul:
r = []
for a in e.args:
if not a.is_bounded:
r.append(a.limit(z, z0, dir))
if r:
return Mul(*r)
r = []
for a in e.args:
r.append(a.limit(z, z0, dir))
elif e.is_Function:
return e.subs(e.args[0], limit(e.args[0], z, z0, dir))
msg = "Don't know how to calculate the limit(%s, %s, %s, dir=%s), sorry."
raise PoleError(msg % (e, z, z0, dir))

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)
obj = Expr.__new__(cls)
obj._args = (e, z, z0, dir)
return obj

def doit(self, **hints):
e, z, z0, dir = self.args
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
return limit(e, z, z0, dir)