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144 lines
4.8 KiB
144 lines
4.8 KiB
from sympy.core.numbers import (Rational, oo, pi)
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.elementary.miscellaneous import (root, sqrt)
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from sympy.functions.elementary.trigonometric import (asin, cos, sin, tan)
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from sympy.polys.rationaltools import together
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from sympy.series.limits import limit
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# Numbers listed with the tests refer to problem numbers in the book
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# "Anti-demidovich, problemas resueltos, Ed. URSS"
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x = Symbol("x")
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def test_leadterm():
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assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
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def root3(x):
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return root(x, 3)
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def root4(x):
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return root(x, 4)
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def test_Limits_simple_0():
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assert limit((2**(x + 1) + 3**(x + 1))/(2**x + 3**x), x, oo) == 3 # 175
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def test_Limits_simple_1():
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assert limit((x + 1)*(x + 2)*(x + 3)/x**3, x, oo) == 1 # 172
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assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0 # 179
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assert limit((2*x - 3)*(3*x + 5)*(4*x - 6)/(3*x**3 + x - 1), x, oo) == 8 # Primjer 1
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assert limit(x/root3(x**3 + 10), x, oo) == 1 # Primjer 2
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assert limit((x + 1)**2/(x**2 + 1), x, oo) == 1 # 181
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def test_Limits_simple_2():
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assert limit(1000*x/(x**2 - 1), x, oo) == 0 # 182
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assert limit((x**2 - 5*x + 1)/(3*x + 7), x, oo) is oo # 183
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assert limit((2*x**2 - x + 3)/(x**3 - 8*x + 5), x, oo) == 0 # 184
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assert limit((2*x**2 - 3*x - 4)/sqrt(x**4 + 1), x, oo) == 2 # 186
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assert limit((2*x + 3)/(x + root3(x)), x, oo) == 2 # 187
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assert limit(x**2/(10 + x*sqrt(x)), x, oo) is oo # 188
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assert limit(root3(x**2 + 1)/(x + 1), x, oo) == 0 # 189
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assert limit(sqrt(x)/sqrt(x + sqrt(x + sqrt(x))), x, oo) == 1 # 190
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def test_Limits_simple_3a():
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a = Symbol('a')
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#issue 3513
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assert together(limit((x**2 - (a + 1)*x + a)/(x**3 - a**3), x, a)) == \
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(a - 1)/(3*a**2) # 196
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def test_Limits_simple_3b():
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h = Symbol("h")
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assert limit(((x + h)**3 - x**3)/h, h, 0) == 3*x**2 # 197
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assert limit((1/(1 - x) - 3/(1 - x**3)), x, 1) == -1 # 198
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assert limit((sqrt(1 + x) - 1)/(root3(1 + x) - 1), x, 0) == Rational(3)/2 # Primer 4
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assert limit((sqrt(x) - 1)/(x - 1), x, 1) == Rational(1)/2 # 199
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assert limit((sqrt(x) - 8)/(root3(x) - 4), x, 64) == 3 # 200
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assert limit((root3(x) - 1)/(root4(x) - 1), x, 1) == Rational(4)/3 # 201
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assert limit(
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(root3(x**2) - 2*root3(x) + 1)/(x - 1)**2, x, 1) == Rational(1)/9 # 202
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def test_Limits_simple_4a():
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a = Symbol('a')
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assert limit((sqrt(x) - sqrt(a))/(x - a), x, a) == 1/(2*sqrt(a)) # Primer 5
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assert limit((sqrt(x) - 1)/(root3(x) - 1), x, 1) == Rational(3, 2) # 205
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assert limit((sqrt(1 + x) - sqrt(1 - x))/x, x, 0) == 1 # 207
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assert limit(sqrt(x**2 - 5*x + 6) - x, x, oo) == Rational(-5, 2) # 213
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def test_limits_simple_4aa():
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assert limit(x*(sqrt(x**2 + 1) - x), x, oo) == Rational(1)/2 # 214
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def test_Limits_simple_4b():
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#issue 3511
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assert limit(x - root3(x**3 - 1), x, oo) == 0 # 215
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def test_Limits_simple_4c():
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assert limit(log(1 + exp(x))/x, x, -oo) == 0 # 267a
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assert limit(log(1 + exp(x))/x, x, oo) == 1 # 267b
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def test_bounded():
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assert limit(sin(x)/x, x, oo) == 0 # 216b
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assert limit(x*sin(1/x), x, 0) == 0 # 227a
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def test_f1a():
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#issue 3508:
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assert limit((sin(2*x)/x)**(1 + x), x, 0) == 2 # Primer 7
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def test_f1a2():
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#issue 3509:
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assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2) # Primer 9
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def test_f1b():
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m = Symbol("m")
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n = Symbol("n")
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h = Symbol("h")
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a = Symbol("a")
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assert limit(sin(x)/x, x, 2) == sin(2)/2 # 216a
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assert limit(sin(3*x)/x, x, 0) == 3 # 217
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assert limit(sin(5*x)/sin(2*x), x, 0) == Rational(5, 2) # 218
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assert limit(sin(pi*x)/sin(3*pi*x), x, 0) == Rational(1, 3) # 219
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assert limit(x*sin(pi/x), x, oo) == pi # 220
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assert limit((1 - cos(x))/x**2, x, 0) == S.Half # 221
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assert limit(x*sin(1/x), x, oo) == 1 # 227b
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assert limit((cos(m*x) - cos(n*x))/x**2, x, 0) == -m**2/2 + n**2/2 # 232
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assert limit((tan(x) - sin(x))/x**3, x, 0) == S.Half # 233
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assert limit((x - sin(2*x))/(x + sin(3*x)), x, 0) == -Rational(1, 4) # 237
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assert limit((1 - sqrt(cos(x)))/x**2, x, 0) == Rational(1, 4) # 239
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assert limit((sqrt(1 + sin(x)) - sqrt(1 - sin(x)))/x, x, 0) == 1 # 240
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assert limit((1 + h/x)**x, x, oo) == exp(h) # Primer 9
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assert limit((sin(x) - sin(a))/(x - a), x, a) == cos(a) # 222, *176
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assert limit((cos(x) - cos(a))/(x - a), x, a) == -sin(a) # 223
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assert limit((sin(x + h) - sin(x))/h, h, 0) == cos(x) # 225
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def test_f2a():
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assert limit(((x + 1)/(2*x + 1))**(x**2), x, oo) == 0 # Primer 8
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def test_f2():
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assert limit((sqrt(
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cos(x)) - root3(cos(x)))/(sin(x)**2), x, 0) == -Rational(1, 12) # *184
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def test_f3():
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a = Symbol('a')
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#issue 3504
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assert limit(asin(a*x)/x, x, 0) == a
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