def test_basic1():
assert limit(x, x, oo) == oo
assert limit(x, x, -oo) == -oo
assert limit(-x, x, oo) == -oo
assert limit(x**2, x, -oo) == oo
assert limit(-x**2, x, oo) == -oo
assert limit(x*log(x), x, 0, dir="+") == 0
assert limit(1/x, x, oo) == 0
assert limit(exp(x), x, oo) == oo
assert limit(-exp(x), x, oo) == -oo
assert limit(exp(x)/x, x, oo) == oo
assert limit(1/x - exp(-x), x, oo) == 0
assert limit(x + 1/x, x, oo) == oo
assert limit(x - x**2, x, oo) == -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == oo
assert limit((1 + x)**oo, x, 0, dir='-') == 0
assert limit((1 + x + y)**oo, x, 0, dir='-') == (1 + y)**(oo)
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
limit(Sum(1/x, (x, 1, y)) - log(y), y, oo)
limit(Sum(1/x, (x, 1, y)) - 1/y, y, oo)
assert limit(gamma(1/x + 3), x, oo) == 2
assert limit(S.NaN, x, -oo) == S.NaN
assert limit(O(2)*x, x, S.NaN) == S.NaN
assert limit(sin(O(x)), x, 0) == 0
assert limit(1/(x - 1), x, 1, dir="+") == oo
assert limit(1/(x - 1), x, 1, dir="-") == -oo
assert limit(1/(5 - x)**3, x, 5, dir="+") == -oo
assert limit(1/(5 - x)**3, x, 5, dir="-") == oo
assert limit(1/sin(x), x, pi, dir="+") == -oo
assert limit(1/sin(x), x, pi, dir="-") == oo
assert limit(1/cos(x), x, pi/2, dir="+") == -oo
assert limit(1/cos(x), x, pi/2, dir="-") == oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="+") == oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="-") == -oo
assert limit(1/cot(x)**3, x, (3*pi/2), dir="+") == -oo
assert limit(1/cot(x)**3, x, (3*pi/2), dir="-") == oo
# approaching 0
# from dir="+"
assert limit(1 + 1/x, x, 0) == oo
# from dir='-'
# Add
assert limit(1 + 1/x, x, 0, dir='-') == -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') == oo
assert limit(x**(-3), x, 0, dir='-') == -oo
assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi))
assert limit((1 + cos(x))**oo, x, 0) == oo
assert limit(x**2, x, 0, dir='real') == 0
assert limit(exp(x), x, 0, dir='real') == 1
pytest.raises(PoleError, lambda: limit(1/x, x, 0, dir='real'))
def test_sympyissue_6068():
assert sqrt(sin(x)).series(x, 0, 8) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 + O(x**8)
assert sqrt(sin(x)).series(x, 0, 10) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 - 67*x**Rational(17, 2)/29030400 + O(x**10)
assert sqrt(sin(x**3)).series(x, 0, 19) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 + O(x**19)
assert sqrt(sin(x**3)).series(x, 0, 20) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 - \
x**Rational(39, 2)/24192 + O(x**20)
def test_sympyissue_12578():
s = root(1 - ((x - 1/x)/2)**(-4), 8)
assert s.series(x, n=17) == (1 - 2*x**4 - 8*x**6 - 34*x**8 -
152*x**10 - 714*x**12 - 3472*x**14 -
17318*x**16 + O(x**17))
d10 = s.diff(x, 10)
assert d10.limit(x, 0) == -551577600
def test_sympyissue_6429():
x = Symbol('x')
c = Symbol('c')
f = (c**2 + x)**(0.5)
assert f.series(x, x0=0, n=1) == (c**2)**0.5 + O(x)
assert f.taylor_term(0, x) == (c**2)**0.5
assert f.taylor_term(1, x) == 0.5*x*(c**2)**(-0.5)
assert f.taylor_term(2, x) == -0.125*x**2*(c**2)**(-1.5)
def test_basic5():
class my(Function):
@classmethod
def eval(cls, arg):
if arg is S.Infinity:
return S.NaN
assert limit(my(x), x, oo) == Limit(my(x), x, oo)
assert limit(4/x > 8, x, 0) # relational test
assert limit(my(x) > 0, x, oo) == Limit(my(x) > 0, x, oo)
# from https://groups.google.com/forum/#!topic/sympy/LkTMQKC_BOw
# fix bisected to ade6d20 and c459d18
a = Symbol('a', positive=True)
f = exp(x*(-a - 1)) * sinh(x)
assert limit(f, x, oo) == 0
assert limit(O(x), x, x**2) == Limit(O(x), x, x**2)
def test_sympyissue_2865():
l1 = limit(O(1/x, (x, oo)), x, 0)
assert l1 != 0 and isinstance(l1, Limit)
l2 = limit(O(x, (x, oo)), x, 0)
assert l2 != 0 and isinstance(l2, Limit)
def test_order_oo():
x = Symbol('x', positive=True, finite=True)
assert O(x)*oo != O(1, x)
assert limit(oo/(x**2 - 4), x, oo) == oo
def test_sympyissue_6653():
x = Symbol('x')
assert (1 / sqrt(1 + sin(x**2))).series(x, 0, 3) == 1 - x**2/2 + O(x**3)
def test_sympyissue_6782():
x = Symbol('x')
assert sqrt(sin(x**3)).series(x, 0, 7) == x**Rational(3, 2) + O(x**7)
assert sqrt(sin(x**4)).series(x, 0, 3) == x**2 + O(x**3)
def test_basic1():
assert limit(x, x, oo) == oo
assert limit(x, x, -oo) == -oo
assert limit(-x, x, oo) == -oo
assert limit(x**2, x, -oo) == oo
assert limit(-x**2, x, oo) == -oo
assert limit(x*log(x), x, 0, dir='+') == 0
assert limit(1/x, x, oo) == 0
assert limit(exp(x), x, oo) == oo
assert limit(-exp(x), x, oo) == -oo
assert limit(exp(x)/x, x, oo) == oo
assert limit(1/x - exp(-x), x, oo) == 0
assert limit(x + 1/x, x, oo) == oo
assert limit(x - x**2, x, oo) == -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == oo
assert limit((1 + x)**oo, x, 0, dir='-') == 0
assert limit((1 + x + y)**oo, x, 0, dir='-') == (1 + y)**oo
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
limit(Sum(1/x, (x, 1, y)) - log(y), y, oo)
limit(Sum(1/x, (x, 1, y)) - 1/y, y, oo)
assert limit(gamma(1/x + 3), x, oo) == 2
assert limit(nan, x, -oo) == nan
assert limit(O(2)*x, x, nan) == nan
assert limit(sin(O(x)), x, 0) == 0
assert limit(1/(x - 1), x, 1, dir='+') == oo
assert limit(1/(x - 1), x, 1, dir='-') == -oo
assert limit(1/(5 - x)**3, x, 5, dir='+') == -oo
assert limit(1/(5 - x)**3, x, 5, dir='-') == oo
assert limit(1/sin(x), x, pi, dir='+') == -oo
assert limit(1/sin(x), x, pi, dir='-') == oo
assert limit(1/cos(x), x, pi/2, dir='+') == -oo
assert limit(1/cos(x), x, pi/2, dir='-') == oo
assert limit(1/tan(x**3), x, cbrt(2*pi), dir='+') == oo
assert limit(1/tan(x**3), x, cbrt(2*pi), dir='-') == -oo
assert limit(1/cot(x)**3, x, 3*pi/2, dir='+') == -oo
assert limit(1/cot(x)**3, x, 3*pi/2, dir='-') == oo
# approaching 0
# from dir='+'
assert limit(1 + 1/x, x, 0) == oo
# from dir='-'
# Add
assert limit(1 + 1/x, x, 0, dir='-') == -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') == oo
assert limit(x**(-3), x, 0, dir='-') == -oo
assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi))
assert limit((1 + cos(x))**oo, x, 0) == oo
assert limit(x**2, x, 0, dir='real') == 0
assert limit(exp(x), x, 0, dir='real') == 1
pytest.raises(PoleError, lambda: limit(1/x, x, 0, dir='real'))
# issue diofant/diofant#74
assert limit(sign(log(1 - 1/x)), x, oo) == -1