8.3
Relative Rates of
Growth
Quick Review
Evaluate the limit
ln x
1. lim
0
e
e
2. lim

x
x
3. lim

e
x
4. lim
0
e
x 
x
x
x 
3
2
x 
2x
2
x 
2x
Find the end behavior model for the function
4
5. f ( x)  -3 x  3 x  x  1
 3x
2x  4x  2
6. f ( x) 
2x 2
x3
4
2
3
e x
7. Let f ( x) 
. Find the
e
(a) local extreme values of f and where they occur.
(b) intervals on which f is increasing. 0, 2
(c) intervals on which f is decreasing.  , 0 and 2, 
 e2  4 
a. Local minimum at 0, 1, Local maximum at  2, 2 
e 

x
2
x
 
What you’ll learn about



Comparing Rates of Growth
Using L’Hôpital’s Rule to Compare Growth Rates
Sequential versus Binary Search
Essential Question
How do we use calculus to understand growth
rates as x→∞ and how it helps us understand the
behavior of functions.
Faster, Slower, Same-rate Growth as x→∞
Let f (x) and g (x) be positive for x sufficiently large,
1. f grows faster than g (and g grows slower than f ) as x → ∞ if
f x 
g x 
lim
 , or, equivalent ly, if lim
 0.
x  g  x 
x  f  x 
2. f and g grow at the same rate as x → ∞ if
f x 
lim
 L  0.
x  g  x 
Example Comparing ex and x3 as x→∞
1. Show that e x grows faster than x 3 as x → ∞.
x
e
This means lim 3  .
x  x

f    e
3
g       

The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.
x
x
x
x
e
e
e
e
lim 3  lim 2  lim

lim
x  3 x
x  x
x  6 x
x  6

Example Comparing ln x with x as x→∞
2. Show that ln x grows slower than x as x → ∞.
ln x




f


ln



This means lim
 0.
x  x
g     
The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.
1
ln x
1/ x
 lim  0
 lim
lim
x  x
x  1
x  x
Example Comparing x with x + sin x as x→∞
3. Show that x grows at the same rate as x + sin x as x → ∞.
x  sin x
f     sin   
This means lim
 L.
x 
g     
x
The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.
x  sin x
 sin x 
lim
 lim 1 
 1
x 
x 
x 
x

Transitivity of Growing Rates
If f grows at the same rate as g as x → ∞ and g grows at the same rate
as h as x → ∞, then f grows at the same rate as h as x → ∞.
Example Growing at the Same Rate as x→∞


4. Show that f x   x  4 and g x   2 x  1
grow at the same rate as x  .
2
2
Show that f and g grow at the same rate by showing that they both grow
at the same rate as h(x) = x.
x 4
x 4
4
 lim

lim
1


1
2
2
x 
x 
x
x
x
2
lim
x 

2
lim
2

2


2
x 1
2 x 1
1 

 lim
 lim  2 
4
2
x 
x 
x 
x
x

x
f x 
 f h  1 1  1
Therefore lim
 lim        
x  g  x 
x  h g

 1 4  4
 
2
Pg. 457, 8.3 #1-37 odd
Example Finding the Order of a Binary Search
For a list of length n, how many steps are required for a binary search?
A binary search takes on the order of log n steps. The reason is if
2
2  n  2 , then m  1  log n  m, and the number of bisections
m -1
m
2
required to narrow the list to one word will be at most m, the smallest
integer greater or equal to log n.
2