Relative
Rates of
Growth
Section 8.2
Comparing Rates of Growth
x
The exponential function e grows so rapidly
and the natural logarithm function ln x grows
so slowly that they set standards by which we
can judge the growth of other functions...
x
As an illustration of how rapidly e grows,
imagine graphing the function on a board
with the axes labeled in centimeters…
e  3 cm high.
6
At x = 6 cm, the graph is e  4 m high.
10
At x = 10 cm, the graph is e  220 m high.
At x = 1 cm, the graph is
1
At x = 24 cm, the graph is more than half way
to the moon.
43
At x = 43 cm, the graph is e  5.0 light-years
high (well past Proxima Centauri, the nearest
star to the Sun).
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
lim

x  g  x 
or, equivalently, if
2. f and g grow at the same rate as
f  x
lim
L0
x  g  x 
g  x
lim
0
x  f  x 
x   if
(L finite and not zero)
Faster, Slower, Same-rate Growth as x 
According to these definitions, y  2 x does not
grow faster than y  x as x   . The two
functions grow at the same rate because
2x
lim
 lim 2  2
x  x
x 
which is a finite nonzero limit. The reason for this
apparent disregard of common sense is that we want
“f grows faster than g” to mean that for large
x-values, g is negligible in comparison to f.
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  .
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the exponential
function as x approaches infinity.
5
 
2
x
5 2

Our new rule: lim
x 
e
x
x
x
 5 
 lim    0
x  2e
 
Grows slower than
(because the base
is less than one!)
e as x  
x
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the exponential
function as x approaches infinity.
x ln x  x
x 1 x   ln x  1
x ln x  x

lim
lim
x
x
x 
x 
e
e
ln x
1x 0
 lim x  lim x   0
x  e
x  e

Grows slower than
e as x  
x
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the squaring function
as x approaches infinity.
x 3
3
x 3
3

lim 2  lim  x  2     0  
x 
x 
x 
x

3
Grows faster than
x as x  
2
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the squaring function
as x approaches infinity.
x  5x
4
x  5x
5
x  5x

lim
1


lim
lim
3
4
2
x

x

x 
x
x
x
 1 0  1
2
Grows at the same rate as x as x  
4
4
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the squaring function
as x approaches infinity.
2
x
ln 2  2
ln 2  2
2


 lim
lim 2  lim
x
x 
x  x
2
2x
2
x
x

 
2
Grows faster than
x as x  
2
x
Guided Practice
Determine whether the given function grows faster than,
at the same rate as, or slower than the natural logarithm
function as x approaches infinity.
log x
ln x   ln10 

log x
log x
 lim
 lim
lim
x 
x  2 ln x
x  ln x
2 ln x
1

2 ln10
Grows at the same rate as ln x as x  
Guided Practice
Show that the three functions grow at the same rate as
x approaches infinity.
f1  x   x
3
x  2x 1
f2  x  
x 1
4
2
2x 1
f3  x   2
x 1
5
Compare the first and second functions:
x  2x 1
4
2
x  2x 1
f2  x 
x

1

lim
 lim
lim
4
3
3
x 
x 
x  f  x 
x x
x
1
1
Rational Function Theorem!   1
1
4
2
Compare the first and third functions:
2 x5  1
5
2
f3  x 
2x 1 2
x

1
 lim
lim
 lim 5


2
3
3
x 
x  f  x 
x  x  x
x
1
1
By transitivity, the second and third functions grow at the
same rate, so all three functions grow at the same rate!