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Limits of Exponential and Logarithmic Functions
Math 130
Supplement to Section 3.1
Exponential Functions
Look at the graph of f ( x)  e x to determine the two basic limits. The first graph shows the
function over the interval [– 2, 4 ]. The next two graph portions show what happens as x increases.
From these we conclude that lim e x  __________ .
x 
In the next series of graphs, the first graph shows f ( x)  e x over the interval [ – 3, 1 ]. The next
two show what happens as x decreases without bound.
lim e x  __________ .
From these we conclude that
Example 1:
x 
Use the two basic limits to find each of the following limits.
a)
lim e 3 x 
b)
c)
lim e 2 x 
d)
e)
lim e x 
f)
x 
x 
3
x 
lim e x 
2
x 
lim e  x 
x 
lim e x 5 
x 
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Logarithmic Functions
Look at the graph of f ( x)  ln x to determine its two basic limits. The first graph shows the
function over the interval [– 1, 6 ]. The next two graphs show what happens as x increases.
From these we conclude that lim ln x  __________ .
x 
In the next series of graphs, the first graph shows f ( x)  ln x over the interval [ – .1, 1 ]. The next
two show what happens as x approaches zero from the right.
From these we conclude that lim ln x  __________ .
x 0
Example 2:
Use the two basic limits to find each of the following limits.
a)
c)
e)
lim ln  4 x  
b)
lim ln  x 4  
d)
lim   ln x  
f)
x 
x 
x  0
lim ln  x 2  5  
x 
lim ln  2 x  
x  0
lim  ln x 1  
x 
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More Examples – Combinations of Functions
Example 3:
Find each of the following limits involving exponentials.
 4x 
lim  x  
x 
e 
 x4 
lim  x  
x  e
 
d)
 x4 
lim   
x  e x
 
lim  x e x  
f)
lim  x 2 e 3 x  
lim  x 2 e 3 x  
h)
lim 3 x e 2 x
lim  x e x  
c)
e)
g)
Example 4:
b)
a)
x 
x 
x 
x 
x 

2

Find each of the following limits involving logarithms.
a)
c)
e)
g)
lim  7  ln x  
b)
lim  4 x  ln x  
d)
lim  x 2  8ln x  
f)
 ln x 
lim 

x 
 x 
h)
x  0
x 
x  0
lim  4 x  ln x  
x 
lim  4 x  ln x  
x  0
lim  x 2 ln x  
x  0
 x 
lim 

x  ln x


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Homework on Limits of Exponential and Logarithmic Functions
Supplement to Section 3.1
Find each of the following limits.
1)
lim  250  e  x  
2)
3)
 x2 
lim 

x  ln x


4)
5)
lim  3x e x  
6)
7)
 x
lim   x
x  e

8)
lim 5 e  x
9)
 ln x 
lim  2  
x 0  x

10)
lim  x 2 e  x  
lim  x 2 e  x  
12)
lim   x e x  
lim   x e  x  
14)
11)
13)
15)
x 
x 



x 
x 
lim  2 x  ln x  
x 
lim  3 x e x  
x 
lim  4 x ln x  
x  0
x 

2

x 
x 
lim  x  ln x  
x  0
lim ln  x  1 
x 1
ANSWERS
1)
250
2)

3)

4)
0
5)

6)
0
7)
0
8)
0
9)
–
10)
0
11)

12)
0
13)

14)

15)
–