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In [1]:

from sympy import *

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In [2]:

x = Symbol("x")

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In [3]:

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

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Out[3]:

2/sqrt(pi)

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

Eqn 8.1

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In [4]:

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

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Out[4]:

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

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In [5]:

limit(_, x, oo)

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Out[5]:

-1

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Eqn 8.2

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In [6]:

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

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Out[6]:

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

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In [7]:

limit(_, x, oo)

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Out[7]:

1

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Eqn 8.3

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In [8]:

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

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Out[8]:

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

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In [9]:

limit(_, x, oo)

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Out[9]:

oo

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Eqn 8.4

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In [10]:

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

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Out[10]:

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

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In [11]:

limit(_, x, oo)

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Out[11]:

-oo

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Eqn 8.5

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In [12]:

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

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Out[12]:

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

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In [13]:

limit(_, x, oo)

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Out[13]:

oo

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Eqn 8.6

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In [14]:

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

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Out[14]:

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

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In [15]:

limit(_, x, oo)

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Out[15]:

oo

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Eqn 8.7

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In [16]:

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

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Out[16]:

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

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In [17]:

limit(_, x, oo)

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Out[17]:

1

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Eqn 8.8

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In [18]:

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

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Out[18]:

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

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In [19]:

limit(_, x, oo)

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Out[19]:

1

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Eqn 8.9

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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)

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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)

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In [21]:

limit(_, x, oo)

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Out[21]:

0

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Eqn 8.10

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In [22]:

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

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Out[22]:

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

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In [23]:

limit(_, x, oo)

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Out[23]:

1/3

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Eqn 8.11

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In [24]:

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

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Out[24]:

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

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In [25]:

limit(_, x, oo)

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Out[25]:

-exp(2)

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Eqn 8.12

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In [26]:

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

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Out[26]:

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

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In [27]:

limit(_, x, oo)

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Out[27]:

5

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Eqn 8.13

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In [28]:

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

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Out[28]:

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

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In [29]:

limit(_, x, oo)

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Out[29]:

oo

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Eqn 8.14

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In [30]:

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

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Out[30]:

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

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In [31]:

limit(_, x, oo)

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Out[31]:

oo

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Eqn 8.15

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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))

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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))

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In [33]:

limit(_, x, oo)

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Out[33]:

oo

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Eqn 8.16

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In [34]:

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

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Out[34]:

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

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In [35]:

limit(_, x, oo)

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Out[35]:

1

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Eqn 8.17

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In [36]:

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

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Out[36]:

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

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In [37]:

limit(_, x, oo)

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Out[37]:

1

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Eqn 8.18

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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

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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)))))

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In [39]:

limit(_, x, oo)

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Out[39]:

2

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Eqn 8.19

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In [40]:

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

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Out[40]:

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

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In [41]:

limit(_, x, oo)

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Out[41]:

1

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Eqn 8.20

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In [42]:

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

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Out[42]:

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

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In [43]:

limit(_, x, oo)

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Out[43]:

E

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

Eqn 8.21

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In [44]:

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

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Out[44]:

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

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In [45]:

limit(_, x, oo)

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Out[45]:

1

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Eqn 8.22

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In [46]:

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

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Out[46]:

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

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In [47]:

limit(_, x, oo)

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Out[47]:

1

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Eqn 8.23

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In [48]:

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

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Out[48]:

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

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In [49]:

limit(_, x, oo)

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Out[49]:

-2/sqrt(pi)

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Eqn 8.24

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In [50]:

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

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Out[50]:

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

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In [51]:

limit(_, x, oo)

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Out[51]:

-1

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Eqn 8.25

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In [52]:

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

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Out[52]:

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

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In [53]:

#limit(_, x, oo)

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Eqn 8.26

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In [54]:

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

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Out[54]:

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

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In [55]:

limit(_, x, oo)

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Out[55]:

oo

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Eqn 8.27

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In [56]:

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

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Out[56]:

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

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In [57]:

#limit(_, x, oo)

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Eqn 8.28

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In [58]:

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

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Out[58]:

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

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In [59]:

limit(_, x, oo)

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Out[59]:

1

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Eqn 8.29

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In [60]:

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

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Out[60]:

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

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In [61]:

limit(_, x, oo)

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Out[61]:

1/2

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Eqn 8.30

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In [62]:

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

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Out[62]:

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

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In [63]:

limit(_, x, oo)

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Out[63]:

-1/2

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Eqn 8.31

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In [64]:

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

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Out[64]:

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

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In [65]:

limit(_, x, oo)

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Out[65]:

oo

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Eqn 8.32

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In [66]:

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

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Out[66]:

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

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In [67]:

limit(_, x, oo)

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Out[67]:

oo

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Eqn 8.33

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In [68]:

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

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Out[68]:

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

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In [69]:

limit(_, x, oo)

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Out[69]:

exp(-1/2)

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Eqn 8.34

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In [70]:

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

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Out[70]:

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

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In [71]:

limit(_, x, oo)

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Out[71]:

exp(-1/2)

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Eqn 8.35

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In [72]:

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

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Out[72]:

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

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In [73]:

limit(_, x, oo)

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Out[73]:

0

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Eqn 8.36

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In [74]:

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

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Out[74]:

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

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In [75]:

#limit(_, x, oo)

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Eqn 8.37

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In [76]:

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

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Out[76]:

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

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In [77]:

#limit(_, x, oo)

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Some other examples

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In [78]:

digamma(digamma(digamma(x)))

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Out[78]:

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

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In [79]:

limit(_, x, oo)

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Out[79]:

oo

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In [80]:

loggamma(loggamma(x))

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Out[80]:

loggamma(loggamma(x))

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In [81]:

limit(_, x, oo)

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Out[81]:

oo

``````