8.3 Relative Rates of Growth

Greg Kelly, Hanford High School, Richland, Washington

The function grows very fast.xy e

We could graph it on the chalkboard:

If x is 3 inches, y is about 20 inches:

3,20

At 10 inches,

1 mile

3

x

y

At 44 inches,

2 million light-years

x

y

We have gone less than We have gone less than half-way across the board half-way across the board horizontally, and already the horizontally, and already the y-value would reach the y-value would reach the Andromeda Galaxy!Andromeda Galaxy!

At 64 inches, the y-value At 64 inches, the y-value would be at the edge of the would be at the edge of the known universe!known universe!(13 billion light-years)(13 billion light-years)

The function grows very slowly.lny xIf 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!

64 inches

By the time we reach the By the time we reach the edge of the universe again edge of the universe again (13 billion light-years) the (13 billion light-years) the chalk line will only have chalk line will only have reached 64 inches!reached 64 inches!

The function increases everywhere, even though it increases extremely slowly.

lny x

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 ifx

limx

f x

g x or

lim 0x

g x

f x

2. f and g grow at the same rate as ifx

lim 0x

f xL

g x

WARNINGWARNINGWARNING

Please temporarily suspend your common sense.

According to this definition, does not grow faster than .

2y xy x

2y x

y x

2lim lim 2 2x x

x

x

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”!

Which grows faster, or ?xe 2x

2lim

x

x

e

x

This is indeterminate, so we apply L’Hôpital’s rule.

lim2

x

x

e

x still

lim2

x

x

e

Still indeterminate.

grows faster than .xe 2x

approaches

We can confirm this graphically:

2

xey

x

“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 and grow at

the same rate as .

2 5f x x 2

2 1g x x

x

Let h x x

2 5limx

x

x

2

2

5limx

x

x

2

2 2

5limx

x

x x 2

5lim 1x x

10

2

2 1limx

x

x

2

2

2 1limx

x

x

2

2 1limx

x

x x

limx

f

g

40

limx

f h

h g

11

4 1

4 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 ifx

lim 0x

f x

g x

We write and say “f is little-oh of g.” of g

Saying is another way to say that f grows slower than g.

of g

Order and Oh-Notation

Saying is another way to say that f grows no faster than g.

Of g

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 if there is a

positive integer M for whichx

f xM

g x

We write and say “f is big-oh of g.” Of g

for x sufficiently large