7/30/2019 Limits of Infinity

1/6

Limits at Infinity

Here we consider limits at infinity using Maple. Recall that the

definition of limx/N

f x = L is that giveneO 0, there is an

MO 0 such that if x > M, then f x L ! e. Graphically,

this means that if x is large enough, the graph off x lies in

the horizontal strip L e, LC e . See the following graph:

x

10 20 30 40

y

0.8

1.0

1.2

1.4

1.6

1.8

2.0

We see that for x bigger than, say, 19, the graph lies entirely in

the blue strip. Moreover, the oscillations seem to (and really

do) dampen out and the graph gets arbitrarily close toy= 1.

We can investigate limits at infinity with the computer or

7/30/2019 Limits of Infinity

2/6

calculator by (1) graphing the function for large x and

inspecting the graph to make a conjecture about the limit, or

(2) using built-in commands to calculate the limit. Of course,

you also need to be able to calculate limits at infinity without

the use of technology.

For the graphical approach, we consider the following

example.

Example 3

For each of the following functions:(i) Plot each function for large vlaues of x.

(ii) Use the plot to conjecture the limitof the function as x goes

toN

(iii) Determine precisely the limit of the function as x goes to

N.

a) e t = 1C t sin tt

b) f x =sin t

t

c)g x =2 x

3C 7 x

x2

Solution:

a) The plot of the first function for large values of t is:

7/30/2019 Limits of Infinity

3/6

x10 20 30 40 50

1

0.5

0

0.5

1

1.5

Clearly (why?) this function does not approach a horizontalline as x increases without bound. Therefore we do not expect

that there is a limit. (And there is none.)Question: Can you

carefully verify that there is no limit?

b) The plot of the function in part b for large values of t is:

7/30/2019 Limits of Infinity

4/6

t

50 100 150

0.4

0.2

0

0.2

0.4

0.6

0.8

We expect that the limit of this function is zero as t increases

without bound. Sincesin t

t%

1

t, and since

limt/N

1

t= 0 (why?), we have, via the sandwich theorem,

limt/N

sin t

t = 0.

c) The plot of the function is, for large x,

7/30/2019 Limits of Infinity

5/6

x

5 10 15 20 25 30

10

20

30

40

50

60

It seems clear that there is no horizontal asymptote here. In

fact, we may write2$x

3C 7$x

x2

=2 x

2C 7

x= 2 xC

7

xand

from the last expression it is easy to see that because2xgrows

without bound as x increases and because7

xgets arbitrarily

small as x increases without bound, the function grows like2x;

i.e. it does not have a limit.

We can calculate limits without appealing to a graph as well.

There is a built-in command "limit" that will calculate just

7/30/2019 Limits of Infinity

6/6

(2)(2)

OO

OO

(1)(1)

about every limit we're interested in. (Though it sometimes has

trouble...) For example, let's calculate limx/N

23$xCx

9xCx

3. All we

need to do is

limit2

3$xCx

9xCx

3, x=N ;

0

As another example we calculate limx/N 1C

5

x

2$x

:

limit 1C5

x

2$x

, x =N ;

e10