Relative Rates of Growth
When we compare functions, one on the properties that we are most interested in is the
rates of growth. The comparisons of exponential, logarithmic, and polynomial
functions can be made precise by defining what it means for a ratio of two function to
grow as x approached infinity.
Growth as x −> ∞
Let f(x) and g(x) be positive for x sufficiently large.
f 
x
, or, equivalently, if
x  g 
x
x   if lim
 f grows faster than g as
g 
x
0.
x f 
x
lim
 f and g grows at the same rate as
x   if lim
not zero).
Example
Compare the growth rates of
x
2
e and x .
Solution:
ex
ex
ex
lim 2 lim
lim 
x  x
x  2 x
x  2
Note: we had to apply L’Hopital’s Rule twice.
Therefore
x
2
e grows faster than x .
Example
Compare the growth rates of ln(x) and
Solution:
1
ln 
x
1
lim 2 lim x lim 2 0
x  x
x 2 x
x 2 x
Therefore ln(x) grows slower than
x2 .
x2 .
x
f 
x
L 0 (L is finite and
g 
x
Example
ln 
x and log10 
x .
Compare the growth rates of
Solution:
1
ln 
x
x
lim
lim
ln 
10.
x log 
x

1
1
10 x
x ln 
10 
Since ln(10) is a finite number, both
ln 
x and log10 
x grow at the same rate.
Growth as “Oh – Notation”
Let f(x) and g(x) be positive for x sufficiently large.
f grows slower than g as
We write
f 
x
0 .
x  g 
x
x   if lim
f o 
g and say “f is little-oh of g”.
Let f(x) and g(x) be positive for x sufficiently large. Then f is of at most the order of
g as
x   if there is a positive integer M for which
large. We write
f O 
g and say “f is big-oh of g”.
Example
Show that
x sin 
x O 
x
2
2
2
as
x  .
Solution:
x 2 sin 
x2
sin 
x2
1  2 1 1 2
x2
x
x 2 sin 
x2
Therefore
2
x2
Which verifies that
x sin 
x O 
x
2
2
2

f 
x
M for x sufficiently
g 
x
Example
Is
1
1 
O  ?
x 2
x 
Solution:
1
x 2 lim x 1 1 , then the answer is yes.
Since lim
x 1
x  x 2
x