sympy/series/limits.py

   1 from sympy.core import S, Add, sympify, Basic
   2 from sympy.core.methods import RelMeths, ArithMeths
   3 from gruntz import gruntz
   4
   5 def limit(e, z, z0, dir="+"):
   6     """
   7     Compute the limit of e(z) at the point z0.
   8
   9     z0 can be any expression, including oo and -oo.
  10
  11     For dir="+" (default) it calculates the limit from the right
  12     (z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0
  13     (oo or -oo), the dir argument doesn't matter.
  14
  15     Examples:
  16
  17     >>> limit(sin(x)/x, x, 0)
  18     1
  19     >>> limit(1/x, x, 0, dir="+")
  20     oo
  21     >>> limit(1/x, x, 0, dir="-")
  22     -oo
  23     >>> limit(1/x, x, oo)
  24     0
  25
  26     Strategy:
  27
  28     First we try some heuristics for easy and frequent cases like "x", "1/x",
  29     "x**2" and similar, so that it's fast. For all other cases, we use the
  30     Gruntz algorithm (see the gruntz() function).
  31     """
  32     e = sympify(e)
  33     z = sympify(z)
  34     z0 = sympify(z0)
  35
  36     if e == z:
  37         return z0
  38
  39     if e.is_Rational:
  40         return e
  41
  42     if e.is_Pow:
  43         if e.args[0] == z:
  44             if e.args[1].is_Rational:
  45                 if e.args[1] > 0:
  46                     return z0**e.args[1]
  47                 else:
  48                     if z0 == 0:
  49                         if dir == "+":
  50                             return S.Infinity
  51                         else:
  52                             return -S.Infinity
  53                     else:
  54                         return z0**e.args[1]
  55             if e.args[1].is_number:
  56                 if e.args[1].evalf() > 0:
  57                     return S.Zero
  58                 else:
  59                     if dir == "+":
  60                         return S.Infinity
  61                     else:
  62                         return -S.Infinity
  63
  64     if e.is_Add:
  65         if e.is_polynomial():
  66             return Add(*[limit(term, z, z0, dir) for term in e.args])
  67         else:
  68             # this is a case like limit(x*y+x*z, z, 2) == x*y+2*x
  69             # but we need to make sure, that the general gruntz() algorithm is
  70             # executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0"
  71             r = e.subs(z, z0)
  72             if r is not S.NaN:
  73                 return r
  74
  75     return gruntz(e, z, z0, dir)
  76
  77 class Limit(Basic, RelMeths, ArithMeths):
  78     """Represents unevaluated limit.
  79
  80     Examples:
  81
  82     >>> Limit(sin(x)/x, x, 0)
  83     Limit(1/x*sin(x), x, 0, dir='+')
  84     >>> Limit(1/x, x, 0, dir="-")
  85     Limit(1/x, x, 0, dir='-')
  86     """
  87
  88     def __new__(cls, e, z, z0, dir="+"):
  89         e = sympify(e)
  90         z = sympify(z)
  91         z0 = sympify(z0)
  92         obj = Basic.__new__(cls)
  93         obj._args = (e, z, z0, dir)
  94         return obj
  95
  96     def doit(self):
  97         e, z, z0, dir = self.args
  98         return limit(e, z, z0, dir)