8.3 Relative Rates of Growth
Greg Kelly, Hanford High School, Richland, Washington
x
y

e
The function
grows very fast.
We could graph it on the chalkboard:
If x is 3 inches, y is about 20 inches:
35
30
25
20
15
10
5
3, 20
0
20
We
have
gonethe
less
than
At 64
inches,
y-value
half-way across the board
would be at the edge of the
horizontally,
and already the
known universe!
y-value
would
reach the
(13 billion
light-years)
Andromeda Galaxy!
40
60
At x  10 inches,
At x  44 inches,
1
y  mile y  2 million light-years
3
80

The function y  ln x grows very slowly.
If we graph it on the chalkboard it
looks like this:
We would have to
move 2.6 miles to
the right before the
line moves a foot
above the x-axis!
35
30
25
20
15
10
5
0
64 inches
By the time we reach the
edge of the universe again
(13 billion light-years) the
chalk line will only have
reached 64 inches!
20
40
60
80
The function y  ln x increases everywhere, even though it
increases extremely slowly.

Definitions: 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
lim
x 
f  x
g  x

or
lim
x 
g  x
f  x
2. f and g grow at the same rate as
lim
x 
f  x
g  x
0
x   if
L0

WARNING
Please temporarily suspend your common sense.

According to this definition,
than y  x .
yx
y  2x
does not grow faster
2x
lim
 lim 2  2
x  x
x 
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
y  2x
1 2 3 4 5 6
Since this is a finite non-zero
limit, the functions grow at
the same rate!
The book says that “f grows
faster than g” means that for
large x values, g is negligible
compared to f.
“Grows faster” is not the same as “has a steeper slope”!

x
2
e
x
Which grows faster, or
?
Weiscan
confirm thisso
graphically:
This
indeterminate,
we apply10L’Hôpital’s rule.

ex 
approaches
lim 2 


x  x

9
x

e 
still
lim



x  2 x


ex
y 2
x
8
7
Still indeterminate.
6
5
ex

lim
x  2
4
3
2
1
e
x
grows faster than
x
2
.
0
1
2
3
4
5
6
7
8
9 10

“Growing at the same rate” is transitive.
In other words, if two functions grow at the same rate
as a third function, then the first two functions grow at
the same rate.

Example 4:


Show that f  x   x  5 and g  x   2 x  1 grow at
2
2
the same rate as x   .
Let h  x   x
x  5  lim
lim
x 
x 
x
2

lim
x 

2 x 1
x
2
x2  5

 lim
x 
x2
x2 5
 lim 2  2
x 
x
x

2 x 1
 x
2
2
f
 f h
1 1
lim
 lim     1  
x  g
x  h g
4 4


50
 lim 1  2  1
x 
x
2
2 x
1 0
 lim 

  4
x 
x
 x
f and g grow at
the same rate.

Order and Oh-Notation
Definition f of Smaller Order than g
Let f and g be positive for x sufficiently large. Then f
is of smaller order than g as x   if
lim
x 
We write
f  x
g  x
0
f  o  g  and say “f is little-oh of g.”
Saying f  o  g  is another way to say that f grows
slower than g.

Order and Oh-Notation
Definition f of at Most the Order of g
Let f and g 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
f  x
g  x
We write
 M for x sufficiently large
f  O  g  and say “f is big-oh of g.”
Saying f  O  g  is another way to say that f grows
no faster than g.
p