def test_gruntz_evaluation_slow():
sskip()
assert gruntz((exp(exp(-x/(1+exp(-x))))*exp(-x/(1+exp(-x/(1+exp(-x)))))
*exp(exp(-x+exp(-x/(1+exp(-x))))))
/ (exp(-x/(1+exp(-x))))**2 - exp(x) + x, x, oo) == 2
assert gruntz(exp(exp(exp(x)/(1-1/x)))
- exp(exp(exp(x)/(1-1/x-log(x)**(-log(x))))), x, oo) == -oo
def test_gruntz_eval_special_slow():
sskip()
assert gruntz(gamma(x+1)/sqrt(2*pi)
- exp(-x)*(x**(x+S(1)/2) + x**(x-S(1)/2)/12), x, oo) == oo
assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0
# XXX This sometimes fails!!!
assert gruntz(exp(gamma(x-exp(-x))*exp(1/x)) - exp(gamma(x)), x, oo) == oo
def test_I():
from sympy.functions import sign as sgn
y = Symbol("y")
assert gruntz(I*x, x, oo) == I*oo
assert gruntz(y*I*x, x, oo) == y*I*oo
assert gruntz(y*3*I*x, x, oo) == y*I*oo
assert gruntz(y*3*sin(I)*x, x, oo).simplify() == sgn(y)*I*oo
def test_gruntz_eval_special_fail():
# TODO exponential integral Ei
assert gruntz(
(Ei(x - exp(-exp(x))) - Ei(x)) *exp(-x)*exp(exp(x))*x, x, oo) == -1
# TODO zeta function series
assert gruntz(
exp((log(2) + 1)*x) * (zeta(x + exp(-x)) - zeta(x)), x, oo) == -log(2)
def test_issue_2073():
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)
assert gruntz(expr.subs(c, m), n, oo) == 1
# fail:
assert gruntz(expr.subs(c, p), n, oo).simplify() == \
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
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)
assert gruntz(expr.subs(c, m), n, oo) == 1
# fail:
assert gruntz(expr.subs(c, p), n, oo).simplify() == (2 ** (p + 1) + r - 1) / (r + 1) ** (p + 1)
def test_intractable():
assert gruntz(1/gamma(x), x, oo) == 0
assert gruntz(1/loggamma(x), x, oo) == 0
assert gruntz(gamma(x)/loggamma(x), x, oo) == oo
assert gruntz(exp(gamma(x))/gamma(x), x, oo) == oo
assert gruntz(gamma(x), x, 3) == 2
assert gruntz(gamma(S(1)/7 + 1/x), x, oo) == gamma(S(1)/7)
assert gruntz(log(x**x)/log(gamma(x)), x, oo) == 1
assert gruntz(log(gamma(gamma(x)))/exp(x), x, oo) == oo
def test_limit2():
assert gruntz(x**x, x, 0, dir="+") == 1
assert gruntz((exp(x) - 1)/x, x, 0) == 1
assert gruntz(1 + 1/x, x, oo) == 1
assert gruntz(-exp(1/x), x, oo) == -1
assert gruntz(x + exp(-x), x, oo) == oo
assert gruntz(x + exp(-x**2), x, oo) == oo
assert gruntz(x + exp(-exp(x)), x, oo) == oo
assert gruntz(13 + 1/x - exp(-x), x, oo) == 13
def test_gruntz_evaluation_slow():
_sskip()
# 8.4
assert gruntz(exp(exp(exp(x) / (1 - 1 / x))) - exp(exp(exp(x) / (1 - 1 / x - log(x) ** (-log(x))))), x, oo) == -oo
# 8.18
assert (
gruntz(
(
exp(exp(-x / (1 + exp(-x))))
* exp(-x / (1 + exp(-x / (1 + exp(-x)))))
* exp(exp(-x + exp(-x / (1 + exp(-x)))))
)
/ (exp(-x / (1 + exp(-x)))) ** 2
- exp(x)
+ x,
x,
oo,
)
== 2
)
def test_issue1091():
assert gruntz(x - gamma(1/x), x, oo) == S.EulerGamma
def test_issue545():
assert gruntz(((x**7 + x + 1)/(2**x + x**2))**(-1/x), x, oo) == 2
def test_aseries_trig():
assert cancel(gruntz(1/log(atan(x)), x, oo)
- 1/(log(pi) + log(S(1)/2))) == 0
assert gruntz(1/acot(x), x, -oo) == -oo
def test_issue1715():
assert gruntz((x + 1)**(1/log(x + 1)), x, oo) == E
def test_limit4():
#issue 364
assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5
#issue 364
assert gruntz((3**(1/x) + 5**(1/x))**x, x, 0) == 5
def test_aseries_trig():
assert cancel(gruntz(1/log(atan(x)), x, oo) \
- 1/(log(pi) + log(S(1)/2))) == 0
assert gruntz(1 / acot(x), x, -oo) == -oo
def test_grunts_eval_special_slow_sometimes_fail():
_sskip()
# XXX This sometimes fails!!!
assert gruntz(exp(gamma(x - exp(-x))*exp(1/x)) - exp(gamma(x)), x, oo) == oo
def test_limit4():
skip("Takes too long")
#issue 364
assert gruntz((3**x+5**x)**(1/x), x, oo) == 5
#issue 364
assert gruntz((3**(1/x)+5**(1/x))**x, x, 0) == 5
def test_limit3():
a = Symbol('a')
assert gruntz(x - log(1 + exp(x)), x, oo) == 0
assert gruntz(x - log(a + exp(x)), x, oo) == 0
assert gruntz(exp(x) / (1 + exp(x)), x, oo) == 1
assert gruntz(exp(x) / (a + exp(x)), x, oo) == 1
def test_gruntz_eval_special():
# Gruntz, p. 126
assert gruntz(
exp(x) * (sin(1 / x + exp(-x)) - sin(1 / x + exp(-x**2))), x, oo) == 1
assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2),
x, oo) == -2 / sqrt(pi)
assert gruntz(
exp(exp(x)) * (exp(sin(1 / x + exp(-exp(x)))) - exp(sin(1 / x))), x,
oo) == 1
assert gruntz(exp(x) * (gamma(x + exp(-x)) - gamma(x)), x, oo) == oo
assert gruntz(exp(exp(digamma(digamma(x)))) / x, x, oo) == exp(-S(1) / 2)
assert gruntz(exp(exp(digamma(log(x)))) / x, x, oo) == exp(-S(1) / 2)
assert gruntz(digamma(digamma(digamma(x))), x, oo) == oo
assert gruntz(loggamma(loggamma(x)), x, oo) == oo
assert gruntz(
((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x) - cos(1 / x)) * x *
log(x), x, oo) == -S(1) / 2
assert gruntz(x * (gamma(x-1/gamma(x)) - gamma(x) + log(x)), x, oo) \
== S(1)/2
assert gruntz((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x), x, oo) == 1
def test_issue545():
assert gruntz(((x**7 + x + 1) / (2**x + x**2))**(-1 / x), x, oo) == 2
def test_exp_log_series():
assert gruntz(x / log(log(x * exp(x))), x, oo) == oo
def test_I():
y = Symbol("y")
assert gruntz(I * x, x, oo) == I * oo
assert gruntz(y * I * x, x, oo) == y * I * oo
assert gruntz(y * 3 * I * x, x, oo) == y * I * oo
assert gruntz(y * 3 * sin(I) * x, x, oo) == y * I * oo
def test_gruntz_eval_special_slow():
_sskip()
assert gruntz(gamma(x + 1)/sqrt(2*pi)
- exp(-x)*(x**(x + S(1)/2) + x**(x - S(1)/2)/12), x, oo) == oo
assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0
def test_gruntz_evaluation():
# Gruntz' thesis pp. 122 to 123
# 8.1
assert gruntz(exp(x)*(exp(1/x - exp(-x)) - exp(1/x)), x, oo) == -1
# 8.2
assert gruntz(exp(x)*(exp(1/x + exp(-x) + exp(-x**2))
- exp(1/x - exp(-exp(x)))), x, oo) == 1
# 8.3
assert gruntz(exp(exp(x - exp(-x))/(1 - 1/x)) - exp(exp(x)), x, oo) == oo
# 8.5
assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(exp(x))), x, oo) == oo
# 8.6
assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(x))))),
x, oo) == oo
# 8.7
assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(exp(x)))))),
x, oo) == 1
# 8.8
assert gruntz(exp(exp(x)) / exp(exp(x - exp(-exp(exp(x))))), x, oo) == 1
# 8.9
assert gruntz(log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2
* exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x),
x, oo) == 0
# 8.10
assert gruntz((x*log(x)*(log(x*exp(x) - x**2))**2)
/ (log(log(x**2 + 2*exp(exp(3*x**3*log(x)))))), x, oo) == S(1)/3
# 8.11
assert gruntz((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1)))) - exp(x))/x,
x, oo) == -exp(2)
# 8.12
assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5
# 8.13
assert gruntz(x/log(x**(log(x**(log(2)/log(x))))), x, oo) == oo
# 8.14
assert gruntz(exp(exp(2*log(x**5 + x)*log(log(x))))
/ exp(exp(10*log(x)*log(log(x)))), x, oo) == oo
# 8.15
assert gruntz(exp(exp(S(5)/2*x**(-S(5)/7) + S(21)/8*x**(S(6)/11)
+ 2*x**(-8) + S(54)/17*x**(S(49)/45) ))**8
/ log(log(-log(S(4)/3*x**(-S(5)/14))))**(S(7)/6), x, oo) == oo
# 8.16
assert gruntz((exp(4*x*exp(-x)/(1/exp(x) + 1/exp(2*x**2/(x + 1)))) - exp(x))
/ exp(x)**4, x, oo) == 1
# 8.17
assert gruntz(exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))/exp(x), x, oo) \
== 1
# 8.19
assert gruntz(log(x)*(log(log(x) + log(log(x))) - log(log(x)))
/ (log(log(x) + log(log(log(x))))), x, oo) == 1
# 8.20
assert gruntz(exp((log(log(x + exp(log(x)*log(log(x))))))
/ (log(log(log(exp(x) + x + log(x)))))), x, oo) == E
# Another
assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(x)), x, oo) == oo
def test_gruntz_evaluation():
# Gruntz' thesis pp. 122 to 123
assert gruntz(exp(x) * (exp(1 / x - exp(-x)) - exp(1 / x)), x, oo) == -1
assert gruntz((x * log(x) * (log(x * exp(x) - x**2))**2) /
(log(log(x**2 + 2 * exp(exp(3 * x**3 * log(x)))))), x,
oo) == S(1) / 3
assert gruntz((3**x + 5**x)**(1 / x), x, oo) == 5
assert gruntz(
exp(
exp(
S(5) / 2 * x**(-S(5) / 7) + S(21) / 8 * x**(S(6) / 11) +
2 * x**(-8) + S(54) / 17 * x**(S(49) / 45)))**8 /
log(log(-log(S(4) / 3 * x**(-S(5) / 14))))**(S(7) / 6), x, oo) == oo
assert gruntz(exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1))))/exp(x), x, oo) \
== 1
assert gruntz(
log(x) * (log(log(x) + log(log(x))) - log(log(x))) /
(log(log(x) + log(log(log(x))))), x, oo) == 1
assert gruntz(x / log(x**(log(x**(log(2) / log(x))))), x, oo) == oo
assert gruntz(x / log(x**(log(x**(log(2) / log(x))))), x, oo) == oo
assert gruntz(
log(x)**2 * exp(
sqrt(log(x)) *
(log(log(x)))**2 * exp(sqrt(log(log(x))) *
(log(log(log(x))))**3)) / sqrt(x), x,
oo) == 0
assert gruntz(
exp((log(log(x + exp(log(x) * log(log(x)))))) /
(log(log(log(exp(x) + x + log(x)))))), x, oo) == E
assert gruntz(exp(x)*(exp(1/x+exp(-x)+exp(-x**2)) \
- exp(1/x-exp(-exp(x)))), x, oo) == 1
assert gruntz(
(exp(4 * x * exp(-x) /
(1 / exp(x) + 1 / exp(2 * x**2 / (x + 1)))) - exp(x)) / exp(x)**4,
x, oo) == 1
assert gruntz(exp(exp(x - exp(-x)) / (1 - 1 / x)) - exp(exp(x)), x,
oo) == oo
assert gruntz(
exp(exp(2 * log(x**5 + x) * log(log(x)))) /
exp(exp(10 * log(x) * log(log(x)))), x, oo) == oo
assert gruntz(
(exp(x * exp(-x) / (exp(-x) + exp(-2 * x**2 / (x + 1)))) - exp(x)) / x,
x, oo) == -exp(2)
assert gruntz(exp(exp(x)) / exp(exp(x - exp(-exp(exp(x))))), x, oo) == 1
assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(exp(x))), x, oo) == oo
assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(exp(x))), x, oo) == oo
assert gruntz(
exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(exp(x)))))), x, oo) == 1
def test_issue1715():
assert gruntz((x + 1)**(1 / log(x + 1)), x, oo) == E
def test_limit1():
assert gruntz(x, x, oo) == oo
assert gruntz(x, x, -oo) == -oo
assert gruntz(-x, x, oo) == -oo
assert gruntz(x**2, x, -oo) == oo
assert gruntz(-x**2, x, oo) == -oo
assert gruntz(x*log(x), x, 0, dir="+") == 0
assert gruntz(1/x, x, oo) == 0
assert gruntz(exp(x), x, oo) == oo
assert gruntz(-exp(x), x, oo) == -oo
assert gruntz(exp(x)/x, x, oo) == oo
assert gruntz(1/x - exp(-x), x, oo) == 0
assert gruntz(x + 1/x, x, oo) == oo
def test_issue_4814():
assert gruntz((x + 1)**(1/log(x + 1)), x, oo) == E
def test_limit3():
a = Symbol('a')
assert gruntz(x - log(1 + exp(x)), x, oo) == 0
assert gruntz(x - log(a + exp(x)), x, oo) == 0
assert gruntz(exp(x)/(1 + exp(x)), x, oo) == 1
assert gruntz(exp(x)/(a + exp(x)), x, oo) == 1
def test_issue_7096():
from sympy.functions import sign
assert gruntz(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi))
def test_I():
y = Symbol("y")
assert gruntz(I*x, x, oo) == I*oo
assert gruntz(y*I*x, x, oo) == y*I*oo
assert gruntz(y*3*I*x, x, oo) == y*I*oo
assert gruntz(y*3*sin(I)*x, x, oo) == y*I*oo
def test_gruntz_evaluation():
# Gruntz' thesis pp. 122 to 123
assert gruntz(exp(x)*(exp(1/x-exp(-x))-exp(1/x)), x, oo) == -1
assert gruntz((x*log(x)*(log(x*exp(x)-x**2))**2)
/ (log(log(x**2+2*exp(exp(3*x**3*log(x)))))), x, oo) == S(1)/3
assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5
assert gruntz(exp(exp(S(5)/2*x**(-S(5)/7)+ S(21)/8*x**(S(6)/11)
+2*x**(-8)+S(54)/17*x**(S(49)/45) ))**8
/ log(log(-log(S(4)/3*x**(-S(5)/14))))**(S(7)/6), x, oo) == oo
assert gruntz(exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1))))/exp(x), x, oo) \
== 1
assert gruntz(log(x)*(log(log(x)+log(log(x))) - log(log(x)))
/ (log(log(x)+log(log(log(x))))), x, oo) == 1
assert gruntz(x/log(x**(log(x**(log(2)/log(x))))), x, oo) == oo
assert gruntz(log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2
* exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x),
x, oo) == 0
assert gruntz(exp((log(log(x+exp(log(x)*log(log(x))))))
/ (log(log(log(exp(x)+x+log(x)))))), x, oo) == E
assert gruntz(exp(x)*(exp(1/x+exp(-x)+exp(-x**2)) \
- exp(1/x-exp(-exp(x)))), x, oo) == 1
assert gruntz((exp(4*x*exp(-x)/(1/exp(x)+1/exp(2*x**2/(x+1)))) - exp(x))
/ exp(x)**4, x, oo) == 1
assert gruntz(exp(exp(x-exp(-x))/(1-1/x)) - exp(exp(x)), x, oo) == oo
assert gruntz(exp(exp(2*log(x**5+x)*log(log(x))))
/ exp(exp(10*log(x)*log(log(x)))), x, oo) == oo
assert gruntz((exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1)))) - exp(x))/x,
x, oo) == -exp(2)
assert gruntz(exp(exp(x)) / exp(exp(x-exp(-exp(exp(x))))), x, oo) == 1
assert gruntz(exp(exp(exp(x+exp(-x)))) / exp(exp(exp(x))), x, oo) == oo
assert gruntz(exp(exp(exp(x+exp(-x)))) / exp(exp(x)), x, oo) == 1
assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x-exp(-exp(exp(x)))))),
x, oo) == 1
def test_limit1():
assert gruntz(x, x, oo) == oo
assert gruntz(x, x, -oo) == -oo
assert gruntz(-x, x, oo) == -oo
assert gruntz(x**2, x, -oo) == oo
assert gruntz(-x**2, x, oo) == -oo
assert gruntz(x * log(x), x, 0, dir="+") == 0
assert gruntz(1 / x, x, oo) == 0
assert gruntz(exp(x), x, oo) == oo
assert gruntz(-exp(x), x, oo) == -oo
assert gruntz(exp(x) / x, x, oo) == oo
assert gruntz(1 / x - exp(-x), x, oo) == 0
assert gruntz(x + 1 / x, x, oo) == oo
def test_issue_1010():
assert gruntz(1/gamma(x), x, 0) == 0
assert gruntz(x*gamma(x), x, 0) == 1
def test_exp_log_series():
assert gruntz(x/log(log(x*exp(x))), x, oo) == oo
def test_grunts_eval_special_slow_sometimes_fail():
sskip()
# XXX This sometimes fails!!!
assert gruntz(exp(gamma(x - exp(-x)) * exp(1 / x)) - exp(gamma(x)), x,
oo) == oo
def test_issue3744():
n = Symbol('n', integer=True, positive=True)
r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
assert gruntz(r, x, 1).simplify() == n/2
def test_issue_3583():
assert gruntz(exp(2*Ei(-x))/x**2, x, 0) == exp(2*EulerGamma)
def test_gruntz_hyperbolic():
assert gruntz(cosh(x), x, oo) == oo
assert gruntz(cosh(x), x, -oo) == oo
assert gruntz(sinh(x), x, oo) == oo
assert gruntz(sinh(x), x, -oo) == -oo
assert gruntz(2*cosh(x)*exp(x), x, oo) == oo
assert gruntz(2*cosh(x)*exp(x), x, -oo) == 1
assert gruntz(2*sinh(x)*exp(x), x, oo) == oo
assert gruntz(2*sinh(x)*exp(x), x, -oo) == -1
assert gruntz(tanh(x), x, oo) == 1
assert gruntz(tanh(x), x, -oo) == -1
assert gruntz(coth(x), x, oo) == 1
assert gruntz(coth(x), x, -oo) == -1
def test_limit4():
#issue 364
assert gruntz((3**x + 5**x)**(1 / x), x, oo) == 5
#issue 364
assert gruntz((3**(1 / x) + 5**(1 / x))**x, x, 0) == 5
def test_gruntz_eval_special():
# Gruntz, p. 126
assert gruntz(exp(x)*(sin(1/x + exp(-x)) - sin(1/x + exp(-x**2))), x, oo) == 1
assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2),
x, oo) == -2/sqrt(pi)
assert gruntz(exp(exp(x)) * (exp(sin(1/x + exp(-exp(x)))) - exp(sin(1/x))),
x, oo) == 1
assert gruntz(exp(x)*(gamma(x + exp(-x)) - gamma(x)), x, oo) == oo
assert gruntz(exp(exp(digamma(digamma(x))))/x, x, oo) == exp(-S(1)/2)
assert gruntz(exp(exp(digamma(log(x))))/x, x, oo) == exp(-S(1)/2)
assert gruntz(digamma(digamma(digamma(x))), x, oo) == oo
assert gruntz(loggamma(loggamma(x)), x, oo) == oo
assert gruntz(((gamma(x + 1/gamma(x)) - gamma(x))/log(x) - cos(1/x))
* x*log(x), x, oo) == -S(1)/2
assert gruntz(x * (gamma(x - 1/gamma(x)) - gamma(x) + log(x)), x, oo) \
== S(1)/2
assert gruntz((gamma(x + 1/gamma(x)) - gamma(x)) / log(x), x, oo) == 1
def test_gruntz_eval_special_slow():
sskip()
assert gruntz(
gamma(x + 1) / sqrt(2 * pi) - exp(-x) *
(x**(x + S(1) / 2) + x**(x - S(1) / 2) / 12), x, oo) == oo
assert gruntz(exp(exp(exp(digamma(digamma(digamma(x)))))) / x, x, oo) == 0