Relative Rates of GrowthSection 8.2

The exponential function grows so rapidlyand the natural logarithm function growsso slowly that they set standards by which wecan judge the growth of other functions...

xeln x

Comparing Rates of Growth

As an illustration of how rapidly grows, imagine graphing the function on a boardwith the axes labeled in centimeters…

xe

At x = 1 cm, the graph is cm high.1 3e

At x = 6 cm, the graph is m high.6 4e

At x = 10 cm, the graph is m high.10 220e

At x = 24 cm, the graph is more than half wayto the moon.

At x = 43 cm, the graph is light-yearshigh (well past Proxima Centauri, the neareststar to the Sun).

43 5.0e

Let f (x) and g(x) be positive for x sufficiently large.

Faster, Slower, Same-rate Growth as x

1. f grows faster than g (and g grows slower than f )as ifx

limx

f x

g x or, equivalently, if

lim 0x

g x

f x

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

lim 0x

f xL

g x (L finite and not zero)

According to these definitions, does notgrow faster than as . The twofunctions grow at the same rate because

Faster, Slower, Same-rate Growth as x 2y x

y x x

2lim lim 2 2x x

x

x

which is a finite nonzero limit. The reason for thisapparent disregard of common sense is that we want“f grows faster than g” to mean that for largex-values, g is negligible in comparison to f.

If f grows at the same rate as g as and ggrows at the same rate as h as , then f growsat the same rate as h as .

Transitivity of Growing Ratesx

x x

Determine whether the given function grows faster than,at the same rate as, or slower than the exponentialfunction as x approaches infinity.

Guided Practice

5

2

x

Our new rule: 5 2

limx

xx e

5lim

2

x

x e

0 (because the baseis less than one!)

Grows slower than as x xe

Determine whether the given function grows faster than,at the same rate as, or slower than the exponentialfunction as x approaches infinity.

Guided Practice

lnx x xln

limxx

x x x

e

Grows slower than as x xe

1 ln 1lim

xx

x x x

e

lnlim

xx

x

e

1lim

xx

x

e

00

Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.

Guided Practice

3 3x 3

2

3limx

x

x

Grows faster than as x 2x

2

3limx

xx

0

Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.

Guided Practice

4 5x x4

2

5limx

x x

x

Grows at the same rate as as x 2x

4

4

5limx

x x

x

3

5lim 1x x

1 0 1

Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.

Guided Practice

2x

2

2lim

x

x x

Grows faster than as x 2x

ln 2 2lim

2

x

x x

2ln 2 2

lim2

x

x

2

Determine whether the given function grows faster than,at the same rate as, or slower than the natural logarithmfunction as x approaches infinity.

Guided Practice

log x

loglim

lnx

x

x

Grows at the same rate as as x ln x

loglim

2lnx

x

x

ln ln10lim

2lnx

x

x

1

2ln10

Show that the three functions grow at the same rate asx approaches infinity.

Guided Practice

31f x x

4 2

2

2 1

1

x xf x

x

5

3 2

2 1

1

xf x

x

Compare the first and second functions:

2

1

limx

f x

f x

4 2

3

2 11lim

x

x xxx

4 2

4 3

2 1limx

x x

x x

1

11

Rational Function Theorem!

Compare the first and third functions:

3

1

limx

f x

f x

5

2

3

2 11lim

x

xx

x

5

5 3

2 1limx

x

x x

2

21

By transitivity, the second and third functions grow at thesame rate, so all three functions grow at the same rate!