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In [1]:

    
from sympy import *



In [2]:

    
x = Symbol("x")



In [3]:

    
limit(exp(x)*exp(x**2)*(erf(x+1/exp(x))-erf(x)), x, oo)









    Out[3]:





2/sqrt(pi)

The examples here show the limit computation on exp-log expressions (from Gruntz' thesis pp. 122 to 123)

Eqn 8.1



In [4]:

    
exp(x)*(exp(1/x-exp(-x))-exp(1/x))









    Out[4]:





(-exp(1/x) + exp(-exp(-x) + 1/x))*exp(x)



In [5]:

    
limit(_, x, oo)









    Out[5]:





-1

Eqn 8.2



In [6]:

    
exp(x)*(exp(1/x+exp(-x)+exp(-x**2)) - exp(1/x-exp(-exp(x))))









    Out[6]:





(-exp(-exp(-exp(x)) + 1/x) + exp(exp(-x**2) + exp(-x) + 1/x))*exp(x)



In [7]:

    
limit(_, x, oo)









    Out[7]:





1

Eqn 8.3



In [8]:

    
exp(exp(x-exp(-x))/(1-1/x)) - exp(exp(x))









    Out[8]:





exp(exp(x - exp(-x))/(1 - 1/x)) - exp(exp(x))



In [9]:

    
limit(_, x, oo)









    Out[9]:





oo

Eqn 8.4



In [10]:

    
exp(exp(exp(x)/(1-1/x))) - exp(exp(exp(x)/(1-1/x-log(x)**(-log(x)))))









    Out[10]:





exp(exp(exp(x)/(1 - 1/x))) - exp(exp(exp(x)/(1 - log(x)**(-log(x)) - 1/x)))



In [11]:

    
limit(_, x, oo)









    Out[11]:





-oo

Eqn 8.5



In [12]:

    
exp(exp(exp(x+exp(-x)))) / exp(exp(exp(x)))









    Out[12]:





exp(-exp(exp(x)))*exp(exp(exp(x + exp(-x))))



In [13]:

    
limit(_, x, oo)









    Out[13]:





oo

Eqn 8.6



In [14]:

    
exp(exp(exp(x))) / exp(exp(exp(x-exp(-exp(x)))))









    Out[14]:





exp(exp(exp(x)))*exp(-exp(exp(x - exp(-exp(x)))))



In [15]:

    
limit(_, x, oo)









    Out[15]:





oo

Eqn 8.7



In [16]:

    
exp(exp(exp(x))) / exp(exp(exp(x-exp(-exp(exp(x))))))









    Out[16]:





exp(exp(exp(x)))*exp(-exp(exp(x - exp(-exp(exp(x))))))



In [17]:

    
limit(_, x, oo)









    Out[17]:





1

Eqn 8.8



In [18]:

    
exp(exp(x)) / exp(exp(x-exp(-exp(exp(x)))))









    Out[18]:





exp(exp(x))*exp(-exp(x - exp(-exp(exp(x)))))



In [19]:

    
limit(_, x, oo)









    Out[19]:





1

Eqn 8.9



In [20]:

    
log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2 * exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x)









    Out[20]:





exp(exp(sqrt(log(log(x)))*log(log(log(x)))**3)*sqrt(log(x))*log(log(x))**2)*log(x)**2/sqrt(x)



In [21]:

    
limit(_, x, oo)









    Out[21]:





0

Eqn 8.10



In [22]:

    
(x*log(x)*(log(x*exp(x)-x**2))**2) / (log(log(x**2+2*exp(exp(3*x**3*log(x))))))









    Out[22]:





x*log(x)*log(-x**2 + x*exp(x))**2/log(log(x**2 + 2*exp(exp(3*x**3*log(x)))))



In [23]:

    
limit(_, x, oo)









    Out[23]:





1/3

Eqn 8.11



In [24]:

    
(exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1)))) - exp(x))/x









    Out[24]:





(-exp(x) + exp(x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x))))/x



In [25]:

    
limit(_, x, oo)









    Out[25]:





-exp(2)

Eqn 8.12



In [26]:

    
(3**x + 5**x)**(1/x)









    Out[26]:





(3**x + 5**x)**(1/x)



In [27]:

    
limit(_, x, oo)









    Out[27]:





5

Eqn 8.13



In [28]:

    
x/log(x**(log(x**(log(2)/log(x)))))









    Out[28]:





x/log(x**log(x**(log(2)/log(x))))



In [29]:

    
limit(_, x, oo)









    Out[29]:





oo

Eqn 8.14



In [30]:

    
exp(exp(2*log(x**5+x)*log(log(x)))) / exp(exp(10*log(x)*log(log(x))))









    Out[30]:





exp(-exp(10*log(x)*log(log(x))))*exp(exp(2*log(x**5 + x)*log(log(x))))



In [31]:

    
limit(_, x, oo)









    Out[31]:





oo

Eqn 8.15



In [32]:

    
4*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 / (9*log(log(-log(S(4)/3*x**(-S(5)/14))))**(S(7)/6))









    Out[32]:





4*exp(8*exp(54*x**(49/45)/17 + 21*x**(6/11)/8 + 2/x**8 + 5/(2*x**(5/7))))/(9*log(log(-log(4/(3*x**(5/14)))))**(7/6))



In [33]:

    
limit(_, x, oo)









    Out[33]:





oo

Eqn 8.16



In [34]:

    
(exp(4*x*exp(-x)/(1/exp(x)+1/exp(2*x**2/(x+1)))) - exp(x)) / exp(x)**4









    Out[34]:





(-exp(x) + exp(4*x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x))))*exp(-4*x)



In [35]:

    
limit(_, x, oo)









    Out[35]:





1

Eqn 8.17



In [36]:

    
exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1))))/exp(x)









    Out[36]:





exp(-x)*exp(x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x)))



In [37]:

    
limit(_, x, oo)









    Out[37]:





1

Eqn 8.18



In [38]:

    
(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









    Out[38]:





x - exp(x) + exp(2*x/(1 + exp(-x)))*exp(-x/(1 + exp(-x/(1 + exp(-x)))))*exp(exp(-x/(1 + exp(-x))))*exp(exp(-x + exp(-x/(1 + exp(-x)))))



In [39]:

    
limit(_, x, oo)









    Out[39]:





2

Eqn 8.19



In [40]:

    
log(x)*(log(log(x)+log(log(x))) - log(log(x))) / (log(log(x)+log(log(log(x)))))









    Out[40]:





(log(log(x) + log(log(x))) - log(log(x)))*log(x)/log(log(x) + log(log(log(x))))



In [41]:

    
limit(_, x, oo)









    Out[41]:





1

Eqn 8.20



In [42]:

    
exp((log(log(x+exp(log(x)*log(log(x)))))) / (log(log(log(exp(x)+x+log(x))))))









    Out[42]:





exp(log(log(x + exp(log(x)*log(log(x)))))/log(log(log(x + exp(x) + log(x)))))



In [43]:

    
limit(_, x, oo)









    Out[43]:





E

The following examples show limit computation on special functions (from Gruntz' thesis p. 126)

Eqn 8.21



In [44]:

    
exp(x)*(sin(1/x+exp(-x))-sin(1/x+exp(-x**2)))









    Out[44]:





(sin(exp(-x) + 1/x) - sin(exp(-x**2) + 1/x))*exp(x)



In [45]:

    
limit(_, x, oo)









    Out[45]:





1

Eqn 8.22



In [46]:

    
exp(exp(x)) * (exp(sin(1/x+exp(-exp(x)))) - exp(sin(1/x)))









    Out[46]:





(-exp(sin(1/x)) + exp(sin(exp(-exp(x)) + 1/x)))*exp(exp(x))



In [47]:

    
limit(_, x, oo)









    Out[47]:





1

Eqn 8.23



In [48]:

    
(erf(x-exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2)









    Out[48]:





(-erf(x) + erf(x - exp(-exp(x))))*exp(x**2)*exp(exp(x))



In [49]:

    
limit(_, x, oo)









    Out[49]:





-2/sqrt(pi)

Eqn 8.24



In [50]:

    
(Ei(x-exp(-exp(x))) - Ei(x)) *exp(-x)*exp(exp(x))*x









    Out[50]:





x*(-Ei(x) + Ei(x - exp(-exp(x))))*exp(-x)*exp(exp(x))



In [51]:

    
limit(_, x, oo)









    Out[51]:





-1

Eqn 8.25



In [52]:

    
exp((log(2)+1)*x) * (zeta(x+exp(-x)) - zeta(x))









    Out[52]:





(-zeta(x) + zeta(x + exp(-x)))*exp(x*(log(2) + 1))



In [53]:

    
#limit(_, x, oo)

Eqn 8.26



In [54]:

    
exp(x)*(gamma(x+exp(-x)) - gamma(x))









    Out[54]:





(-gamma(x) + gamma(x + exp(-x)))*exp(x)



In [55]:

    
limit(_, x, oo)









    Out[55]:





oo

Eqn 8.27



In [56]:

    
exp(gamma(x-exp(-x))*exp(1/x)) - exp(gamma(x))









    Out[56]:





exp(exp(1/x)*gamma(x - exp(-x))) - exp(gamma(x))



In [57]:

    
#limit(_, x, oo)

Eqn 8.28



In [58]:

    
(gamma(x+1/gamma(x)) - gamma(x)) / log(x)









    Out[58]:





(-gamma(x) + gamma(x + 1/gamma(x)))/log(x)



In [59]:

    
limit(_, x, oo)









    Out[59]:





1

Eqn 8.29



In [60]:

    
x * (gamma(x-1/gamma(x)) - gamma(x) + log(x))









    Out[60]:





x*(log(x) - gamma(x) + gamma(x - 1/gamma(x)))



In [61]:

    
limit(_, x, oo)









    Out[61]:





1/2

Eqn 8.30



In [62]:

    
((gamma(x+1/gamma(x)) - gamma(x))/log(x) - cos(1/x))*x*log(x)









    Out[62]:





x*((-gamma(x) + gamma(x + 1/gamma(x)))/log(x) - cos(1/x))*log(x)



In [63]:

    
limit(_, x, oo)









    Out[63]:





-1/2

Eqn 8.31



In [64]:

    
gamma(x+1)/sqrt(2*pi) - exp(-x)*(x**(x+S(1)/2) + x**(x-S(1)/2)/12)









    Out[64]:





-(x**(x - 1/2)/12 + x**(x + 1/2))*exp(-x) + sqrt(2)*gamma(x + 1)/(2*sqrt(pi))



In [65]:

    
limit(_, x, oo)









    Out[65]:





oo

Eqn 8.32



In [66]:

    
log(gamma(gamma(x)))/exp(x)









    Out[66]:





exp(-x)*log(gamma(gamma(x)))



In [67]:

    
limit(_, x, oo)









    Out[67]:





oo

Eqn 8.33



In [68]:

    
exp(exp(digamma(digamma(x))))/x









    Out[68]:





exp(exp(polygamma(0, polygamma(0, x))))/x



In [69]:

    
limit(_, x, oo)









    Out[69]:





exp(-1/2)

Eqn 8.34



In [70]:

    
exp(exp(digamma(log(x))))/x









    Out[70]:





exp(exp(polygamma(0, log(x))))/x



In [71]:

    
limit(_, x, oo)









    Out[71]:





exp(-1/2)

Eqn 8.35



In [72]:

    
exp(exp(exp(digamma(digamma(digamma(x))))))/x









    Out[72]:





exp(exp(exp(polygamma(0, polygamma(0, polygamma(0, x))))))/x



In [73]:

    
limit(_, x, oo)









    Out[73]:





0

Eqn 8.36



In [74]:

    
besselj(2,x)*exp(x*(2*log(2+sqrt(3))-sqrt(3)))*sqrt(x)









    Out[74]:





sqrt(x)*exp(x*(-sqrt(3) + 2*log(sqrt(3) + 2)))*besselj(2, x)



In [75]:

    
#limit(_, x, oo)

Eqn 8.37



In [76]:

    
Max(x, exp(x))/log(Min(exp(-x), exp(-exp(x))))









    Out[76]:





Max(x, exp(x))/log(Min(exp(-x), exp(-exp(x))))



In [77]:

    
#limit(_, x, oo)

Some other examples



In [78]:

    
digamma(digamma(digamma(x)))









    Out[78]:





polygamma(0, polygamma(0, polygamma(0, x)))



In [79]:

    
limit(_, x, oo)









    Out[79]:





oo



In [80]:

    
loggamma(loggamma(x))









    Out[80]:





loggamma(loggamma(x))



In [81]:

    
limit(_, x, oo)









    Out[81]:





oo