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5.2-5.3 Study Guide
NO CALCULATOR
What is the limit as x approaches infinity of the following:
a) e  x
c) e x
b) log( x)
d) ln x
What is the limit as x approaches 0 from the right of the following:
a) log( x)
b) ln x
Write the functions polynomial, log, and exponent in order from largest infinity to smallest
infinity:
Find each limit.
 x 
1. lim   x 
x  e



2  x2
4. lim x e
x 
2. lim  x  ln x 
3. lim  ln( x 1) 
5. lim   ln x 
 5 x10 
6. lim  3 x 
x  4e


x 0

x 0
x 1
e2 x  2e x  3
x 0
ex 1
3x  4 x
x
5x
7. lim
7
9. lim
x 5  3 x
8. lim
10. lim(ln 4 x  ln 2 x)
x 
9  5e x
11. lim
x 0 x  6
Use this space to write out all of the exponential and logarithmic derivatives, if necessary.
(Try not to peek at your notes!)
Differentiate each function. Simplify completely.
1. f ( x)  x  ln x
2. f ( x)  x 2 ln x
3. y 
ln x
x
5. f ( x)  x2e x
4. y 
2
7. y  ln( x 2  5x  1)
9. f ( x) 
ex
x 1
ex
x4
6. f ( x)  ln( x 4 )
 x2  1 
8. y  ln 

 x 
10. y  3e2 x
2
Find the critical points, and determine where each function is increasing or decreasing.
1. f ( x)  x3e2 x
2. f ( x) 
3ln 2 x
(Numbers you need: e2/2=3.6945 ln((2)(1))=.693 ln((2)(5))=2.079)
x