from itertools import product as cartes

from sympy import (
    limit, exp, oo, log, sqrt, Limit, sin, floor, cos, ceiling, sinh,
    atan, Abs, gamma, Symbol, S, pi, Integral, Rational, I, E, besselj,
    tan, cot, integrate, Sum, sign, Function, subfactorial, symbols,
    binomial, simplify, frac, Float, sec, zoo, fresnelc, fresnels, real_root,
    acos, erf, erfc, erfi, LambertW, factorial, digamma, uppergamma, re,
    Ei, EulerGamma, asin, atanh, acot, acoth, asec, acsc, cbrt, besselk)

from sympy.calculus.util import AccumBounds
from sympy.core.mul import Mul
from sympy.series.limits import heuristics
from sympy.series.order import Order
from sympy.testing.pytest import XFAIL, raises

from sympy.abc import x, y, z, k
n = Symbol('n', integer=True, positive=True)


def test_basic1():
    assert limit(x, x, oo) is oo
    assert limit(x, x, -oo) is -oo
    assert limit(-x, x, oo) is -oo
    assert limit(x**2, x, -oo) is oo
    assert limit(-x**2, x, oo) is -oo
    assert limit(x*log(x), x, 0, dir="+") == 0
    assert limit(1/x, x, oo) == 0
    assert limit(exp(x), x, oo) is oo
    assert limit(-exp(x), x, oo) is -oo
    assert limit(exp(x)/x, x, oo) is oo
    assert limit(1/x - exp(-x), x, oo) == 0
    assert limit(x + 1/x, x, oo) is oo
    assert limit(x - x**2, x, oo) is -oo
    assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
    assert limit((1 + x)**oo, x, 0) == Limit((x + 1)**oo, x, 0)
    assert limit((1 + x)**oo, x, 0, dir='-') == Limit((x + 1)**oo, x, 0, dir='-')
    assert limit((1 + x + y)**oo, x, 0, dir='-') == Limit((1 + x + y)**oo, x, 0, dir='-')
    assert limit(y/x/log(x), x, 0) == -oo*sign(y)
    assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
    assert limit(gamma(1/x + 3), x, oo) == 2
    assert limit(S.NaN, x, -oo) is S.NaN
    assert limit(Order(2)*x, x, S.NaN) is S.NaN
    assert limit(1/(x - 1), x, 1, dir="+") is oo
    assert limit(1/(x - 1), x, 1, dir="-") is -oo
    assert limit(1/(5 - x)**3, x, 5, dir="+") is -oo
    assert limit(1/(5 - x)**3, x, 5, dir="-") is oo
    assert limit(1/sin(x), x, pi, dir="+") is -oo
    assert limit(1/sin(x), x, pi, dir="-") is oo
    assert limit(1/cos(x), x, pi/2, dir="+") is -oo
    assert limit(1/cos(x), x, pi/2, dir="-") is oo
    assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="+") is oo
    assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="-") is -oo
    assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="+") is -oo
    assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="-") is oo

    # test bi-directional limits
    assert limit(sin(x)/x, x, 0, dir="+-") == 1
    assert limit(x**2, x, 0, dir="+-") == 0
    assert limit(1/x**2, x, 0, dir="+-") is oo

    # test failing bi-directional limits
    assert limit(1/x, x, 0, dir="+-") is zoo
    # approaching 0
    # from dir="+"
    assert limit(1 + 1/x, x, 0) is oo
    # from dir='-'
    # Add
    assert limit(1 + 1/x, x, 0, dir='-') is -oo
    # Pow
    assert limit(x**(-2), x, 0, dir='-') is oo
    assert limit(x**(-3), x, 0, dir='-') is -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*(-1)**(1 - pi)
    assert limit((1 + cos(x))**oo, x, 0) == Limit((cos(x) + 1)**oo, x, 0)


def test_basic2():
    assert limit(x**x, x, 0, dir="+") == 1
    assert limit((exp(x) - 1)/x, x, 0) == 1
    assert limit(1 + 1/x, x, oo) == 1
    assert limit(-exp(1/x), x, oo) == -1
    assert limit(x + exp(-x), x, oo) is oo
    assert limit(x + exp(-x**2), x, oo) is oo
    assert limit(x + exp(-exp(x)), x, oo) is oo
    assert limit(13 + 1/x - exp(-x), x, oo) == 13


def test_basic3():
    assert limit(1/x, x, 0, dir="+") is oo
    assert limit(1/x, x, 0, dir="-") is -oo


def test_basic4():
    assert limit(2*x + y*x, x, 0) == 0
    assert limit(2*x + y*x, x, 1) == 2 + y
    assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8
    assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0
    assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9


def test_log():
    # https://github.com/sympy/sympy/issues/21598
    a, b, c = symbols('a b c', positive=True)
    A = log(a/b) - (log(a) - log(b))
    assert A.limit(a, oo) == 0
    assert (A * c).limit(a, oo) == 0

    tau, x = symbols('tau x', positive=True)
    # The value of manualintegrate in the issue
    expr = tau**2*((tau - 1)*(tau + 1)*log(x + 1)/(tau**2 + 1)**2 + 1/((tau**2\
            + 1)*(x + 1)) - (-2*tau*atan(x/tau) + (tau**2/2 - 1/2)*log(tau**2\
            + x**2))/(tau**2 + 1)**2)
    assert limit(expr, x, oo) == pi*tau**3/(tau**2 + 1)**2


def test_piecewise():
    # https://github.com/sympy/sympy/issues/18363
    assert limit((real_root(x - 6, 3) + 2)/(x + 2), x, -2, '+') == Rational(1, 12)


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)


def test_issue_3885():
    assert limit(x*y + x*z, z, 2) == x*y + 2*x


def test_Limit():
    assert Limit(sin(x)/x, x, 0) != 1
    assert Limit(sin(x)/x, x, 0).doit() == 1
    assert Limit(x, x, 0, dir='+-').args == (x, x, 0, Symbol('+-'))


def test_floor():
    assert limit(floor(x), x, -2, "+") == -2
    assert limit(floor(x), x, -2, "-") == -3
    assert limit(floor(x), x, -1, "+") == -1
    assert limit(floor(x), x, -1, "-") == -2
    assert limit(floor(x), x, 0, "+") == 0
    assert limit(floor(x), x, 0, "-") == -1
    assert limit(floor(x), x, 1, "+") == 1
    assert limit(floor(x), x, 1, "-") == 0
    assert limit(floor(x), x, 2, "+") == 2
    assert limit(floor(x), x, 2, "-") == 1
    assert limit(floor(x), x, 248, "+") == 248
    assert limit(floor(x), x, 248, "-") == 247

    # https://github.com/sympy/sympy/issues/14478
    assert limit(x*floor(3/x)/2, x, 0, '+') == Rational(3, 2)
    assert limit(floor(x + 1/2) - floor(x), x, oo) == AccumBounds(-0.5, 1.5)


def test_floor_requires_robust_assumptions():
    assert limit(floor(sin(x)), x, 0, "+") == 0
    assert limit(floor(sin(x)), x, 0, "-") == -1
    assert limit(floor(cos(x)), x, 0, "+") == 0
    assert limit(floor(cos(x)), x, 0, "-") == 0
    assert limit(floor(5 + sin(x)), x, 0, "+") == 5
    assert limit(floor(5 + sin(x)), x, 0, "-") == 4
    assert limit(floor(5 + cos(x)), x, 0, "+") == 5
    assert limit(floor(5 + cos(x)), x, 0, "-") == 5


def test_ceiling():
    assert limit(ceiling(x), x, -2, "+") == -1
    assert limit(ceiling(x), x, -2, "-") == -2
    assert limit(ceiling(x), x, -1, "+") == 0
    assert limit(ceiling(x), x, -1, "-") == -1
    assert limit(ceiling(x), x, 0, "+") == 1
    assert limit(ceiling(x), x, 0, "-") == 0
    assert limit(ceiling(x), x, 1, "+") == 2
    assert limit(ceiling(x), x, 1, "-") == 1
    assert limit(ceiling(x), x, 2, "+") == 3
    assert limit(ceiling(x), x, 2, "-") == 2
    assert limit(ceiling(x), x, 248, "+") == 249
    assert limit(ceiling(x), x, 248, "-") == 248

    # https://github.com/sympy/sympy/issues/14478
    assert limit(x*ceiling(3/x)/2, x, 0, '+') == Rational(3, 2)
    assert limit(ceiling(x + 1/2) - ceiling(x), x, oo) == AccumBounds(-0.5, 1.5)


def test_ceiling_requires_robust_assumptions():
    assert limit(ceiling(sin(x)), x, 0, "+") == 1
    assert limit(ceiling(sin(x)), x, 0, "-") == 0
    assert limit(ceiling(cos(x)), x, 0, "+") == 1
    assert limit(ceiling(cos(x)), x, 0, "-") == 1
    assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
    assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
    assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
    assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6


def test_atan():
    x = Symbol("x", real=True)
    assert limit(atan(x)*sin(1/x), x, 0) == 0
    assert limit(atan(x) + sqrt(x + 1) - sqrt(x), x, oo) == pi/2


def test_set_signs():
    assert limit(abs(x), x, 0) == 0
    assert limit(abs(sin(x)), x, 0) == 0
    assert limit(abs(cos(x)), x, 0) == 1
    assert limit(abs(sin(x + 1)), x, 0) == sin(1)

    # https://github.com/sympy/sympy/issues/9449
    assert limit((Abs(x + y) - Abs(x - y))/(2*x), x, 0) == sign(y)

    # https://github.com/sympy/sympy/issues/12398
    assert limit(Abs(log(x)/x**3), x, oo) == 0
    assert limit(x*(Abs(log(x)/x**3)/Abs(log(x + 1)/(x + 1)**3) - 1), x, oo) == 3

    # https://github.com/sympy/sympy/issues/18501
    assert limit(Abs(log(x - 1)**3 - 1), x, 1, '+') == oo

    # https://github.com/sympy/sympy/issues/18997
    assert limit(Abs(log(x)), x, 0) == oo
    assert limit(Abs(log(Abs(x))), x, 0) == oo

    # https://github.com/sympy/sympy/issues/19026
    z = Symbol('z', positive=True)
    assert limit(Abs(log(z) + 1)/log(z), z, oo) == 1

    # https://github.com/sympy/sympy/issues/20704
    assert limit(z*(Abs(1/z + y) - Abs(y - 1/z))/2, z, 0) == 0

    # https://github.com/sympy/sympy/issues/21606
    assert limit(cos(z)/sign(z), z, pi, '-') == -1


def test_heuristic():
    x = Symbol("x", real=True)
    assert heuristics(sin(1/x) + atan(x), x, 0, '+') == AccumBounds(-1, 1)
    assert limit(log(2 + sqrt(atan(x))*sqrt(sin(1/x))), x, 0) == log(2)


def test_issue_3871():
    z = Symbol("z", positive=True)
    f = -1/z*exp(-z*x)
    assert limit(f, x, oo) == 0
    assert f.limit(x, oo) == 0


def test_exponential():
    n = Symbol('n')
    x = Symbol('x', real=True)
    assert limit((1 + x/n)**n, n, oo) == exp(x)
    assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2)
    assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2)
    assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2)
    assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + S.Exp1
    assert limit((2 + 6*x)**x/(6*x)**x, x, oo) == exp(S('1/3'))


def test_exponential2():
    n = Symbol('n')
    assert limit((1 + x/(n + sin(n)))**n, n, oo) == exp(x)


def test_doit():
    f = Integral(2 * x, x)
    l = Limit(f, x, oo)
    assert l.doit() is oo


def test_series_AccumBounds():
    assert limit(sin(k) - sin(k + 1), k, oo) == AccumBounds(-2, 2)
    assert limit(cos(k) - cos(k + 1) + 1, k, oo) == AccumBounds(-1, 3)

    # not the exact bound
    assert limit(sin(k) - sin(k)*cos(k), k, oo) == AccumBounds(-2, 2)

    # test for issue #9934
    t1 = Mul(AccumBounds(-S(3)/2 + cos(1)/2, cos(1)/2 + S.Half), 1/(-1 + cos(1)))
    assert limit(simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t1

    t2 = Mul(AccumBounds(-1 + sin(1)/2, sin(1)/2 + 1), 1/(1 - cos(1)))
    assert limit(simplify(Sum(sin(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t2

    assert limit(frac(x)**x, x, oo) == AccumBounds(0, oo)  # wolfram gives (0, 1)
    assert limit(((sin(x) + 1)/2)**x, x, oo) == AccumBounds(0, oo)  # wolfram says 0


@XFAIL
def test_doit2():
    f = Integral(2 * x, x)
    l = Limit(f, x, oo)
    # limit() breaks on the contained Integral.
    assert l.doit(deep=False) == l


def test_issue_2929():
    assert limit((x * exp(x))/(exp(x) - 1), x, -oo) == 0


def test_issue_3792():
    assert limit((1 - cos(x))/x**2, x, S.Half) == 4 - 4*cos(S.Half)
    assert limit(sin(sin(x + 1) + 1), x, 0) == sin(1 + sin(1))
    assert limit(abs(sin(x + 1) + 1), x, 0) == 1 + sin(1)


def test_issue_4090():
    assert limit(1/(x + 3), x, 2) == Rational(1, 5)
    assert limit(1/(x + pi), x, 2) == S.One/(2 + pi)
    assert limit(log(x)/(x**2 + 3), x, 2) == log(2)/7
    assert limit(log(x)/(x**2 + pi), x, 2) == log(2)/(4 + pi)


def test_issue_4547():
    assert limit(cot(x), x, 0, dir='+') is oo
    assert limit(cot(x), x, pi/2, dir='+') == 0


def test_issue_5164():
    assert limit(x**0.5, x, oo) == oo**0.5 is oo
    assert limit(x**0.5, x, 16) == S(16)**0.5
    assert limit(x**0.5, x, 0) == 0
    assert limit(x**(-0.5), x, oo) == 0
    assert limit(x**(-0.5), x, 4) == S(4)**(-0.5)


def test_issue_5383():
    func = (1.0 * 1 + 1.0 * x)**(1.0 * 1 / x)
    assert limit(func, x, 0) == E.n()


def test_issue_14793():
    expr = ((x + S(1)/2) * log(x) - x + log(2*pi)/2 - \
        log(factorial(x)) + S(1)/(12*x))*x**3
    assert limit(expr, x, oo) == S(1)/360


def test_issue_5183():
    # using list(...) so py.test can recalculate values
    tests = list(cartes([x, -x],
                        [-1, 1],
                        [2, 3, S.Half, Rational(2, 3)],
                        ['-', '+']))
    results = (oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), oo,
               0, 0, 0, 0, 0, 0, 0, 0,
               oo, oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3),
               0, 0, 0, 0, 0, 0, 0, 0)
    assert len(tests) == len(results)
    for i, (args, res) in enumerate(zip(tests, results)):
        y, s, e, d = args
        eq = y**(s*e)
        try:
            assert limit(eq, x, 0, dir=d) == res
        except AssertionError:
            if 0:  # change to 1 if you want to see the failing tests
                print()
                print(i, res, eq, d, limit(eq, x, 0, dir=d))
            else:
                assert None


def test_issue_5184():
    assert limit(sin(x)/x, x, oo) == 0
    assert limit(atan(x), x, oo) == pi/2
    assert limit(gamma(x), x, oo) is oo
    assert limit(cos(x)/x, x, oo) == 0
    assert limit(gamma(x), x, S.Half) == sqrt(pi)

    r = Symbol('r', real=True)
    assert limit(r*sin(1/r), r, 0) == 0


def test_issue_5229():
    assert limit((1 + y)**(1/y) - S.Exp1, y, 0) == 0


def test_issue_4546():
    # using list(...) so py.test can recalculate values
    tests = list(cartes([cot, tan],
                        [-pi/2, 0, pi/2, pi, pi*Rational(3, 2)],
                        ['-', '+']))
    results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0,
               oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo)
    assert len(tests) == len(results)
    for i, (args, res) in enumerate(zip(tests, results)):
        f, l, d = args
        eq = f(x)
        try:
            assert limit(eq, x, l, dir=d) == res
        except AssertionError:
            if 0:  # change to 1 if you want to see the failing tests
                print()
                print(i, res, eq, l, d, limit(eq, x, l, dir=d))
            else:
                assert None


def test_issue_3934():
    assert limit((1 + x**log(3))**(1/x), x, 0) == 1
    assert limit((5**(1/x) + 3**(1/x))**x, x, 0) == 5


def test_calculate_series():
    # needs gruntz calculate_series to go to n = 32
    assert limit(x**Rational(77, 3)/(1 + x**Rational(77, 3)), x, oo) == 1
    # needs gruntz calculate_series to go to n = 128
    assert limit(x**101.1/(1 + x**101.1), x, oo) == 1


def test_issue_5955():
    assert limit((x**16)/(1 + x**16), x, oo) == 1
    assert limit((x**100)/(1 + x**100), x, oo) == 1
    assert limit((x**1885)/(1 + x**1885), x, oo) == 1
    assert limit((x**1000/((x + 1)**1000 + exp(-x))), x, oo) == 1


def test_newissue():
    assert limit(exp(1/sin(x))/exp(cot(x)), x, 0) == 1


def test_extended_real_line():
    assert limit(x - oo, x, oo) == Limit(x - oo, x, oo)
    assert limit(1/(x + sin(x)) - oo, x, 0) == Limit(1/(x + sin(x)) - oo, x, 0)
    assert limit(oo/x, x, oo) == Limit(oo/x, x, oo)
    assert limit(x - oo + 1/x, x, oo) == Limit(x - oo + 1/x, x, oo)


@XFAIL
def test_order_oo():
    x = Symbol('x', positive=True)
    assert Order(x)*oo != Order(1, x)
    assert limit(oo/(x**2 - 4), x, oo) is oo


def test_issue_5436():
    raises(NotImplementedError, lambda: limit(exp(x*y), x, oo))
    raises(NotImplementedError, lambda: limit(exp(-x*y), x, oo))


def test_Limit_dir():
    raises(TypeError, lambda: Limit(x, x, 0, dir=0))
    raises(ValueError, lambda: Limit(x, x, 0, dir='0'))


def test_polynomial():
    assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, oo) == 1
    assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, -oo) == 1


def test_rational():
    assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, oo) == (z - 1)/(y*z)
    assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, -oo) == (z - 1)/(y*z)


def test_issue_5740():
    assert limit(log(x)*z - log(2*x)*y, x, 0) == oo*sign(y - z)


def test_issue_6366():
    n = Symbol('n', integer=True, positive=True)
    r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
    assert limit(r, x, 1).cancel() == n/2


def test_factorial():
    from sympy import factorial, E
    f = factorial(x)
    assert limit(f, x, oo) is oo
    assert limit(x/f, x, oo) == 0
    # see Stirling's approximation:
    # https://en.wikipedia.org/wiki/Stirling's_approximation
    assert limit(f/(sqrt(2*pi*x)*(x/E)**x), x, oo) == 1
    assert limit(f, x, -oo) == factorial(-oo)


def test_issue_6560():
    e = (5*x**3/4 - x*Rational(3, 4) + (y*(3*x**2/2 - S.Half) +
                             35*x**4/8 - 15*x**2/4 + Rational(3, 8))/(2*(y + 1)))
    assert limit(e, y, oo) == 5*x**3/4 + 3*x**2/4 - 3*x/4 - Rational(1, 4)

@XFAIL
def test_issue_5172():
    n = Symbol('n')
    r = Symbol('r', positive=True)
    c = Symbol('c')
    p = Symbol('p', positive=True)
    m = Symbol('m', negative=True)
    expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c +
            (r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n)
    expr = expr.subs(c, c + 1)
    raises(NotImplementedError, lambda: limit(expr, n, oo))
    assert limit(expr.subs(c, m), n, oo) == 1
    assert limit(expr.subs(c, p), n, oo).simplify() == \
        (2**(p + 1) + r - 1)/(r + 1)**(p + 1)


def test_issue_7088():
    a = Symbol('a')
    assert limit(sqrt(x/(x + a)), x, oo) == 1


def test_branch_cuts():
    assert limit(asin(I*x + 2), x, 0) == pi - asin(2)
    assert limit(asin(I*x + 2), x, 0, '-') == asin(2)
    assert limit(asin(I*x - 2), x, 0) == -asin(2)
    assert limit(asin(I*x - 2), x, 0, '-') == -pi + asin(2)
    assert limit(acos(I*x + 2), x, 0) == -acos(2)
    assert limit(acos(I*x + 2), x, 0, '-') == acos(2)
    assert limit(acos(I*x - 2), x, 0) == acos(-2)
    assert limit(acos(I*x - 2), x, 0, '-') == 2*pi - acos(-2)
    assert limit(atan(x + 2*I), x, 0) == I*atanh(2)
    assert limit(atan(x + 2*I), x, 0, '-') == -pi + I*atanh(2)
    assert limit(atan(x - 2*I), x, 0) == pi - I*atanh(2)
    assert limit(atan(x - 2*I), x, 0, '-') == -I*atanh(2)
    assert limit(atan(1/x), x, 0) == pi/2
    assert limit(atan(1/x), x, 0, '-') == -pi/2
    assert limit(atan(x), x, oo) == pi/2
    assert limit(atan(x), x, -oo) == -pi/2
    assert limit(acot(x + S(1)/2*I), x, 0) == pi - I*acoth(S(1)/2)
    assert limit(acot(x + S(1)/2*I), x, 0, '-') == -I*acoth(S(1)/2)
    assert limit(acot(x - S(1)/2*I), x, 0) == I*acoth(S(1)/2)
    assert limit(acot(x - S(1)/2*I), x, 0, '-') == -pi + I*acoth(S(1)/2)
    assert limit(acot(x), x, 0) == pi/2
    assert limit(acot(x), x, 0, '-') == -pi/2
    assert limit(asec(I*x + S(1)/2), x, 0) == asec(S(1)/2)
    assert limit(asec(I*x + S(1)/2), x, 0, '-') == -asec(S(1)/2)
    assert limit(asec(I*x - S(1)/2), x, 0) == 2*pi - asec(-S(1)/2)
    assert limit(asec(I*x - S(1)/2), x, 0, '-') == asec(-S(1)/2)
    assert limit(acsc(I*x + S(1)/2), x, 0) == acsc(S(1)/2)
    assert limit(acsc(I*x + S(1)/2), x, 0, '-') == pi - acsc(S(1)/2)
    assert limit(acsc(I*x - S(1)/2), x, 0) == -pi + acsc(S(1)/2)
    assert limit(acsc(I*x - S(1)/2), x, 0, '-') == -acsc(S(1)/2)

    assert limit(log(I*x - 1), x, 0) == I*pi
    assert limit(log(I*x - 1), x, 0, '-') == -I*pi
    assert limit(log(-I*x - 1), x, 0) == -I*pi
    assert limit(log(-I*x - 1), x, 0, '-') == I*pi

    assert limit(sqrt(I*x - 1), x, 0) == I
    assert limit(sqrt(I*x - 1), x, 0, '-') == -I
    assert limit(sqrt(-I*x - 1), x, 0) == -I
    assert limit(sqrt(-I*x - 1), x, 0, '-') == I

    assert limit(cbrt(I*x - 1), x, 0) == (-1)**(S(1)/3)
    assert limit(cbrt(I*x - 1), x, 0, '-') == -(-1)**(S(2)/3)
    assert limit(cbrt(-I*x - 1), x, 0) == -(-1)**(S(2)/3)
    assert limit(cbrt(-I*x - 1), x, 0, '-') == (-1)**(S(1)/3)


def test_issue_6364():
    a = Symbol('a')
    e = z/(1 - sqrt(1 + z)*sin(a)**2 - sqrt(1 - z)*cos(a)**2)
    assert limit(e, z, 0) == 1/(cos(a)**2 - S.Half)


def test_issue_4099():
    a = Symbol('a')
    assert limit(a/x, x, 0) == oo*sign(a)
    assert limit(-a/x, x, 0) == -oo*sign(a)
    assert limit(-a*x, x, oo) == -oo*sign(a)
    assert limit(a*x, x, oo) == oo*sign(a)


def test_issue_4503():
    dx = Symbol('dx')
    assert limit((sqrt(1 + exp(x + dx)) - sqrt(1 + exp(x)))/dx, dx, 0) == \
        exp(x)/(2*sqrt(exp(x) + 1))


def test_issue_8208():
    assert limit(n**(Rational(1, 1e9) - 1), n, oo) == 0


def test_issue_8229():
    assert limit((x**Rational(1, 4) - 2)/(sqrt(x) - 4)**Rational(2, 3), x, 16) == 0


def test_issue_8433():
    d, t = symbols('d t', positive=True)
    assert limit(erf(1 - t/d), t, oo) == -1


def test_issue_8481():
    k = Symbol('k', integer=True, nonnegative=True)
    lamda = Symbol('lamda', real=True, positive=True)
    limit(lamda**k * exp(-lamda) / factorial(k), k, oo) == 0


def test_issue_8730():
    assert limit(subfactorial(x), x, oo) is oo


def test_issue_9252():
    n = Symbol('n', integer=True)
    c = Symbol('c', positive=True)
    assert limit((log(n))**(n/log(n)) / (1 + c)**n, n, oo) == 0
    # limit should depend on the value of c
    raises(NotImplementedError, lambda: limit((log(n))**(n/log(n)) / c**n, n, oo))


def test_issue_9558():
    assert limit(sin(x)**15, x, 0, '-') == 0


def test_issue_10801():
    # make sure limits work with binomial
    assert limit(16**k / (k * binomial(2*k, k)**2), k, oo) == pi


def test_issue_10976():
    s, x = symbols('s x', real=True)
    assert limit(erf(s*x)/erf(s), s, 0) == x


def test_issue_9041():
    assert limit(factorial(n) / ((n/exp(1))**n * sqrt(2*pi*n)), n, oo) == 1


def test_issue_9205():
    x, y, a = symbols('x, y, a')
    assert Limit(x, x, a).free_symbols == {a}
    assert Limit(x, x, a, '-').free_symbols == {a}
    assert Limit(x + y, x + y, a).free_symbols == {a}
    assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a}


def test_issue_9471():
    assert limit(((27**(log(n,3)))/n**3),n,oo) == 1
    assert limit(((27**(log(n,3)+1))/n**3),n,oo) == 27


def test_issue_11496():
    assert limit(erfc(log(1/x)), x, oo) == 2


def test_issue_11879():
    assert simplify(limit(((x+y)**n-x**n)/y, y, 0)) == n*x**(n-1)


def test_limit_with_Float():
    k = symbols("k")
    assert limit(1.0 ** k, k, oo) == 1
    assert limit(0.3*1.0**k, k, oo) == Float(0.3)


def test_issue_10610():
    assert limit(3**x*3**(-x - 1)*(x + 1)**2/x**2, x, oo) == Rational(1, 3)


def test_issue_6599():
    assert limit((n + cos(n))/n, n, oo) == 1


def test_issue_12555():
    assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, -oo) == 2
    assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, oo) is oo


def test_issue_12769():
    r, z, x = symbols('r z x', real=True)
    a, b, s0, K, F0, s, T = symbols('a b s0 K F0 s T', positive=True, real=True)
    fx = (F0**b*K**b*r*s0 - sqrt((F0**2*K**(2*b)*a**2*(b - 1) + \
        F0**(2*b)*K**2*a**2*(b - 1) + F0**(2*b)*K**(2*b)*s0**2*(b - 1)*(b**2 - 2*b + 1) - \
        2*F0**(2*b)*K**(b + 1)*a*r*s0*(b**2 - 2*b +  1) + \
        2*F0**(b + 1)*K**(2*b)*a*r*s0*(b**2 - 2*b + 1) - \
        2*F0**(b + 1)*K**(b + 1)*a**2*(b - 1))/((b - 1)*(b**2 - 2*b + 1))))*(b*r -  b - r + 1)

    assert fx.subs(K, F0).factor(deep=True) == limit(fx, K, F0).factor(deep=True)


def test_issue_13332():
    assert limit(sqrt(30)*5**(-5*x - 1)*(46656*x)**x*(5*x + 2)**(5*x + 5*S.Half) *
                (6*x + 2)**(-6*x - 5*S.Half), x, oo) == Rational(25, 36)


def test_issue_12564():
    assert limit(x**2 + x*sin(x) + cos(x), x, -oo) is oo
    assert limit(x**2 + x*sin(x) + cos(x), x, oo) is oo
    assert limit(((x + cos(x))**2).expand(), x, oo) is oo
    assert limit(((x + sin(x))**2).expand(), x, oo) is oo
    assert limit(((x + cos(x))**2).expand(), x, -oo) is oo
    assert limit(((x + sin(x))**2).expand(), x, -oo) is oo


def test_issue_14456():
    raises(NotImplementedError, lambda: Limit(exp(x), x, zoo).doit())
    raises(NotImplementedError, lambda: Limit(x**2/(x+1), x, zoo).doit())


def test_issue_14411():
    assert limit(3*sec(4*pi*x - x/3), x, 3*pi/(24*pi - 2)) is -oo


def test_issue_13382():
    assert limit(x*(((x + 1)**2 + 1)/(x**2 + 1) - 1), x, oo) == 2


def test_issue_13403():
    assert limit(x*(-1 + (x + log(x + 1) + 1)/(x + log(x))), x ,oo) == 1


def test_issue_13416():
    assert limit((-x**3*log(x)**3 + (x - 1)*(x + 1)**2*log(x + 1)**3)/(x**2*log(x)**3), x ,oo) == 1


def test_issue_13462():
    assert limit(n**2*(2*n*(-(1 - 1/(2*n))**x + 1) - x - (-x**2/4 + x/4)/n), n, oo) == x**3/24 - x**2/8 + x/12


def test_issue_13750():
    a = Symbol('a')
    assert limit(erf(a - x), x, oo) == -1
    assert limit(erf(sqrt(x) - x), x, oo) == -1


def test_issue_14514():
    assert limit((1/(log(x)**log(x)))**(1/x), x, oo) == 1


def test_issue_14574():
    assert limit(sqrt(x)*cos(x - x**2) / (x + 1), x, oo) == 0


def test_issue_10102():
    assert limit(fresnels(x), x, oo) == S.Half
    assert limit(3 + fresnels(x), x, oo) == 3 + S.Half
    assert limit(5*fresnels(x), x, oo) == Rational(5, 2)
    assert limit(fresnelc(x), x, oo) == S.Half
    assert limit(fresnels(x), x, -oo) == Rational(-1, 2)
    assert limit(4*fresnelc(x), x, -oo) == -2


def test_issue_14377():
    raises(NotImplementedError, lambda: limit(exp(I*x)*sin(pi*x), x, oo))


def test_issue_15146():
    e = (x/2) * (-2*x**3 - 2*(x**3 - 1) * x**2 * digamma(x**3 + 1) + \
        2*(x**3 - 1) * x**2 * digamma(x**3 + x + 1) + x + 3)
    assert limit(e, x, oo) == S(1)/3


def test_issue_15202():
    e = (2**x*(2 + 2**(-x)*(-2*2**x + x + 2))/(x + 1))**(x + 1)
    assert limit(e, x, oo) == exp(1)

    e = (log(x, 2)**7 + 10*x*factorial(x) + 5**x) / (factorial(x + 1) + 3*factorial(x) + 10**x)
    assert limit(e, x, oo) == 10


def test_issue_15282():
    assert limit((x**2000 - (x + 1)**2000) / x**1999, x, oo) == -2000


def test_issue_15984():
    assert limit((-x + log(exp(x) + 1))/x, x, oo, dir='-') == 0


def test_issue_13571():
    assert limit(uppergamma(x, 1) / gamma(x), x, oo) == 1


def test_issue_13575():
    assert limit(acos(erfi(x)), x, 1) == acos(erfi(S.One))


def test_issue_17325():
    assert Limit(sin(x)/x, x, 0, dir="+-").doit() == 1
    assert Limit(x**2, x, 0, dir="+-").doit() == 0
    assert Limit(1/x**2, x, 0, dir="+-").doit() is oo
    assert Limit(1/x, x, 0, dir="+-").doit() is zoo


def test_issue_10978():
    assert LambertW(x).limit(x, 0) == 0


def test_issue_14313_comment():
    assert limit(floor(n/2), n, oo) is oo


@XFAIL
def test_issue_15323():
    d = ((1 - 1/x)**x).diff(x)
    assert limit(d, x, 1, dir='+') == 1


def test_issue_12571():
    assert limit(-LambertW(-log(x))/log(x), x, 1) == 1


def test_issue_14590():
    assert limit((x**3*((x + 1)/x)**x)/((x + 1)*(x + 2)*(x + 3)), x, oo) == exp(1)


def test_issue_14393():
    a, b = symbols('a b')
    assert limit((x**b - y**b)/(x**a - y**a), x, y) == b*y**(-a + b)/a


def test_issue_14556():
    assert limit(factorial(n + 1)**(1/(n + 1)) - factorial(n)**(1/n), n, oo) == exp(-1)


def test_issue_14811():
    assert limit(((1 + ((S(2)/3)**(x + 1)))**(2**x))/(2**((S(4)/3)**(x - 1))), x, oo) == oo


def test_issue_14874():
    assert limit(besselk(0, x), x, oo) == 0


def test_issue_16222():
    assert limit(exp(x), x, 1000000000) == exp(1000000000)


def test_issue_16714():
    assert limit(((x**(x + 1) + (x + 1)**x) / x**(x + 1))**x, x, oo) == exp(exp(1))


def test_issue_16722():
    z = symbols('z', positive=True)
    assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1)
    z = symbols('z', positive=True, integer=True)
    assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1)


def test_issue_17431():
    assert limit(((n + 1) + 1) / (((n + 1) + 2) * factorial(n + 1)) *
                 (n + 2) * factorial(n) / (n + 1), n, oo) == 0
    assert limit((n + 2)**2*factorial(n)/((n + 1)*(n + 3)*factorial(n + 1))
                 , n, oo) == 0
    assert limit((n + 1) * factorial(n) / (n * factorial(n + 1)), n, oo) == 0


def test_issue_17671():
    assert limit(Ei(-log(x)) - log(log(x))/x, x, 1) == EulerGamma


def test_issue_17751():
    a, b, c, x = symbols('a b c x', positive=True)
    assert limit((a + 1)*x - sqrt((a + 1)**2*x**2 + b*x + c), x, oo) == -b/(2*a + 2)


def test_issue_17792():
    assert limit(factorial(n)/sqrt(n)*(exp(1)/n)**n, n, oo) == sqrt(2)*sqrt(pi)


def test_issue_18118():
    assert limit(sign(sin(x)), x, 0, "-") == -1
    assert limit(sign(sin(x)), x, 0, "+") == 1


def test_issue_18306():
    assert limit(sin(sqrt(x))/sqrt(sin(x)), x, 0, '+') == 1


def test_issue_18378():
    assert limit(log(exp(3*x) + x)/log(exp(x) + x**100), x, oo) == 3


def test_issue_18399():
    assert limit((1 - S(1)/2*x)**(3*x), x, oo) is zoo
    assert limit((-x)**x, x, oo) is zoo


def test_issue_18442():
    assert limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-') == Limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-')


def test_issue_18452():
    assert limit(abs(log(x))**x, x, 0) == 1
    assert limit(abs(log(x))**x, x, 0, "-") == 1


def test_issue_18482():
    assert limit((2*exp(3*x)/(exp(2*x) + 1))**(1/x), x, oo) == exp(1)


def test_issue_18508():
    assert limit(sin(x)/sqrt(1-cos(x)), x, 0) == sqrt(2)
    assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='+') == sqrt(2)
    assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='-') == -sqrt(2)


def test_issue_18969():
    a, b = symbols('a b', positive=True)
    assert limit(LambertW(a), a, b) == LambertW(b)
    assert limit(exp(LambertW(a)), a, b) == exp(LambertW(b))


def test_issue_18992():
    assert limit(n/(factorial(n)**(1/n)), n, oo) == exp(1)


def test_issue_19067():
    x = Symbol('x')
    assert limit(gamma(x)/(gamma(x - 1)*gamma(x + 2)), x, 0) == -1


def test_issue_19586():
    assert limit(x**(2**x*3**(-x)), x, oo) == 1


def test_issue_13715():
    n = Symbol('n')
    p = Symbol('p', zero=True)
    assert limit(n + p, n, 0) == 0


def test_issue_15055():
    assert limit(n**3*((-n - 1)*sin(1/n) + (n + 2)*sin(1/(n + 1)))/(-n + 1), n, oo) == 1


def test_issue_16708():
    m, vi = symbols('m vi', positive=True)
    B, ti, d = symbols('B ti d')
    assert limit((B*ti*vi - sqrt(m)*sqrt(-2*B*d*vi + m*(vi)**2) + m*vi)/(B*vi), B, 0) == (d + ti*vi)/vi


def test_issue_19739():
    assert limit((-S(1)/4)**x, x, oo) == 0


def test_issue_19766():
    assert limit(2**(-x)*sqrt(4**(x + 1) + 1), x, oo) == 2


def test_issue_19770():
    m = Symbol('m')
    # the result is not 0 for non-real m
    assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
    m = Symbol('m', real=True)
    # can be improved to give the correct result 0
    assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
    m = Symbol('m', nonzero=True)
    assert limit(cos(m*x), x, oo) == AccumBounds(-1, 1)
    assert limit(cos(m*x)/x, x, oo) == 0


def test_issue_7535():
    assert limit(tan(x)/sin(tan(x)), x, pi/2) == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+')
    assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='-')
    assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-')
    assert limit(sin(tan(x)),x,pi/2) == AccumBounds(-1, 1)
    assert -oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
    assert oo*(1/sin(oo)) == AccumBounds(-oo, oo)
    assert oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
    assert -oo*(1/sin(oo)) == AccumBounds(-oo, oo)


def test_issue_20365():
    assert limit(((x + 1)**(1/x) - E)/x, x, 0) == -E/2


def test_issue_21031():
    assert limit(((1 + x)**(1/x) - (1 + 2*x)**(1/(2*x)))/asin(x), x, 0) == E/2


def test_issue_21038():
    assert limit(sin(pi*x)/(3*x - 12), x, 4) == pi/3


def test_issue_20578():
    expr = abs(x) * sin(1/x)
    assert limit(expr,x,0,'+') == 0
    assert limit(expr,x,0,'-') == 0
    assert limit(expr,x,0,'+-') == 0


def test_issue_21415():
    exp = (x-1)*cos(1/(x-1))
    assert exp.limit(x,1) == 0
    assert exp.expand().limit(x,1) == 0


def test_issue_21530():
    assert limit(sinh(n + 1)/sinh(n), n, oo) == E


def test_issue_21550():
    r = (sqrt(5) - 1)/2
    assert limit((x - r)/(x**2 + x - 1), x, r) == sqrt(5)/5


def test_issue_21661():
    out = limit((x**(x + 1) * (log(x) + 1) + 1) / x, x, 11)
    assert out == S(3138428376722)/11 + 285311670611*log(11)


def test_issue_21701():
    assert limit((besselj(z, x)/x**z).subs(z, 7), x, 0) == S(1)/645120


def test_issue_21721():
    a = Symbol('a', real=True)
    I = integrate(1/(pi*(1 + (x - a)**2)), x)
    assert I.limit(x, oo) == S.Half


def test_issue_21756():
    term = (1 - exp(-2*I*pi*z))/(1 - exp(-2*I*pi*z/5))
    assert term.limit(z, 0) == 5
    assert re(term).limit(z, 0) == 5


def test_issue_21785():
    a = Symbol('a')
    assert sqrt((-a**2 + x**2)/(1 - x**2)).limit(a, 1, '-') == I