sympy/series/tests/test_limits.py

   1 from sympy import limit, exp, oo, log, sqrt, Limit, sin, floor, cos, ceiling, \
   2         atan, Symbol
   3 from sympy.abc import x, y, z
   4 from sympy.utilities.pytest import XFAIL
   5
   6 def test_basic1():
   7     assert limit(x, x, oo) == oo
   8     assert limit(x, x, -oo) == -oo
   9     assert limit(-x, x, oo) == -oo
  10     assert limit(x**2, x, -oo) == oo
  11     assert limit(-x**2, x, oo) == -oo
  12     assert limit(x*log(x), x, 0, dir="+") == 0
  13     assert limit(1/x,x,oo) == 0
  14     assert limit(exp(x),x,oo) == oo
  15     assert limit(-exp(x),x,oo) == -oo
  16     assert limit(exp(x)/x,x,oo) == oo
  17     assert limit(1/x-exp(-x),x,oo) == 0
  18     assert limit(x+1/x,x,oo) == oo
  19
  20
  21 def test_basic2():
  22     assert limit(x**x, x, 0, dir="+") == 1
  23     assert limit((exp(x)-1)/x, x, 0) == 1
  24     assert limit(1+1/x,x,oo) == 1
  25     assert limit(-exp(1/x),x,oo) == -1
  26     assert limit(x+exp(-x),x,oo) == oo
  27     assert limit(x+exp(-x**2),x,oo) == oo
  28     assert limit(x+exp(-exp(x)),x,oo) == oo
  29     assert limit(13+1/x-exp(-x),x,oo) == 13
  30
  31 def test_basic3():
  32     assert limit(1/x, x, 0, dir="+") == oo
  33     assert limit(1/x, x, 0, dir="-") == -oo
  34
  35 def test_basic3():
  36     assert limit(2*x + y*x, x, 0) == 0
  37     assert limit(2*x + y*x, x, 1) == 2+y
  38     assert limit(2*x**8 + y*x**(-3), x, -2) == 512-y/8
  39     assert limit(sqrt(x+1)-sqrt(x),x,oo)==0
  40
  41 def test_issue786():
  42     assert limit(x*y+x*z, z, 2) == x*y+2*x
  43
  44 def test_Limit():
  45     assert Limit(sin(x)/x, x, 0) != 1
  46     assert Limit(sin(x)/x, x, 0).doit() == 1
  47
  48 def test_floor():
  49     assert limit(floor(x), x, -2, "+") == -2
  50     assert limit(floor(x), x, -2, "-") == -3
  51     assert limit(floor(x), x, -1, "+") == -1
  52     assert limit(floor(x), x, -1, "-") == -2
  53     assert limit(floor(x), x, 0, "+") == 0
  54     assert limit(floor(x), x, 0, "-") == -1
  55     assert limit(floor(x), x, 1, "+") == 1
  56     assert limit(floor(x), x, 1, "-") == 0
  57     assert limit(floor(x), x, 2, "+") == 2
  58     assert limit(floor(x), x, 2, "-") == 1
  59     assert limit(floor(x), x, 248, "+") == 248
  60     assert limit(floor(x), x, 248, "-") == 247
  61
  62     # note: if any of the tests below fails, just comment it out. General fix
  63     # needs better assumptions handling.
  64
  65     # this doesn't work, it requires robust assumptions:
  66     assert limit(floor(sin(x)), x, 0, "+") == 0
  67     assert limit(floor(sin(x)), x, 0, "-") == -1
  68     assert limit(floor(cos(x)), x, 0, "+") == 0
  69     assert limit(floor(cos(x)), x, 0, "-") == 0
  70
  71     # this doesn't work, it requires robust assumptions:
  72     assert limit(floor(5+sin(x)), x, 0, "+") == 5
  73     #assert limit(floor(5+sin(x)), x, 0, "-") == 4
  74     #assert limit(floor(5+cos(x)), x, 0, "+") == 5
  75     #assert limit(floor(5+cos(x)), x, 0, "-") == 5
  76
  77 def test_ceiling():
  78     assert limit(ceiling(x), x, -2, "+") == -1
  79     assert limit(ceiling(x), x, -2, "-") == -2
  80     assert limit(ceiling(x), x, -1, "+") == 0
  81     assert limit(ceiling(x), x, -1, "-") == -1
  82     assert limit(ceiling(x), x, 0, "+") == 1
  83     assert limit(ceiling(x), x, 0, "-") == 0
  84     assert limit(ceiling(x), x, 1, "+") == 2
  85     assert limit(ceiling(x), x, 1, "-") == 1
  86     assert limit(ceiling(x), x, 2, "+") == 3
  87     assert limit(ceiling(x), x, 2, "-") == 2
  88     assert limit(ceiling(x), x, 248, "+") == 249
  89     assert limit(ceiling(x), x, 248, "-") == 248
  90
  91     # note: if any of the tests below fails, just comment it out. General fix
  92     # needs better assumptions handling.
  93
  94     # this doesn't work, it requires robust assumptions:
  95     #assert limit(ceiling(sin(x)), x, 0, "+") == 1
  96     assert limit(ceiling(sin(x)), x, 0, "-") == 0
  97     assert limit(ceiling(cos(x)), x, 0, "+") == 1
  98     assert limit(ceiling(cos(x)), x, 0, "-") == 1
  99
 100     # this doesn't work, it requires robust assumptions:
 101     #assert limit(ceiling(5+sin(x)), x, 0, "+") == 6
 102     assert limit(ceiling(5+sin(x)), x, 0, "-") == 5
 103     assert limit(ceiling(5+cos(x)), x, 0, "+") == 6
 104     assert limit(ceiling(5+cos(x)), x, 0, "-") == 6
 105
 106 def test_atan():
 107     x = Symbol("x", real=True)
 108     assert limit(atan(x)*sin(1/x), x, 0) == 0
 109
 110 def test_abs():
 111     assert limit(abs(x), x, 0) == 0
 112     assert limit(abs(sin(x)), x, 0) == 0
 113     assert limit(abs(cos(x)), x, 0) == 1
 114
 115 def test_heuristic():
 116     x = Symbol("x", real=True)
 117     assert limit(log(2+sqrt(atan(x)*sin(1/x))), x, 0) == log(2)
 118
 119 def test_issue772():
 120     f = -1/z*exp(-z*x)
 121     assert limit(f, x, oo) == 0
 122     assert f.limit(x, oo) == 0
 123
 124 def test_exponential():
 125     n = Symbol('n')
 126     assert limit((1+x/n)**n,n,oo) == exp(x)
 127     assert limit((1+x/(2*n))**n,n,oo) == exp(x/2)
 128     assert limit((1+x/(2*n+1))**n,n,oo) == exp(x/2)
 129     assert limit(((x-1)/(x+1))**x,x,oo) == exp(-2)
 130
 131 @XFAIL
 132 def test_exponential2():
 133     n = Symbol('n')
 134     assert limit((1+x/(n+sin(n)))**n,n,oo) == exp(x)