1

Infinite Limits & Vertical Asymptotes

Section 1.5

2

After this lesson, you will be able to:

determine infinite limits from the left and from the right

find and sketch the vertical asymptotes of the graph of a function

3

Infinite Limits-Graphically

0

1limx x

______

0

1limx x

______

Consider the function,

_________ is not in the domain of the function.

1( )f x

x

So, we are curious about what is happening to the function as x approaches 0. We’ll say that as x approaches 0 is a limit of special interest.

Graph the function…

As x approaches 0 from the right, graphically we can see that the function is going off to positive infinity…As x approaches 0 from the left, graphically we can see that the function is going off to negative infinity…

4

Infinite Limits-Numerically

Let’s examine the table of values to reinforce our decision…Set TBL to be .01.

You can see that as x gets closer to 0 from values that are

•larger than 0, the function goes off to positive infinity. (RHL)

•smaller than 0, the function goes off to negative infinity. (LHL)

5

Infinite LimitsEventually, we should be able to find the limit as x approaches 0 without using a table or a graph of the function,

1( )f x

x

First, let’s consider what happens as x approaches 0 from the right (values larger than 0). Evaluate by hand:

We say that the graph _______________ _______________ ___________ as x approaches 0 from the right.

(0.1)

(0.01)

(0.001)

f

f

f

6

Infinite LimitsNext, let’s consider what happens as x approaches 0 from the left (values smaller than 0). Evaluate by hand:

We say that the graph _______________ _______________ ___________ as x approaches 0 from the left.

( 0.1)

( 0.01)

( 0.001)

f

f

f

7

Infinite LimitsYou probably won’t see this definition in the textbook.

1Really large positive # ( + )

Really small positive #

1Really large negative # ( )

Really small negative #

8

Infinite LimitsNo matter how we evaluate the limit (graphically, numerically, or analytically), we know that

0

1limx x

0

1limx x

9

Infinite LimitsIf lim ( ) OR lim ( ) ,

x c x cf x or f x or

the line x = c is a vertical asymptote to the graph.

x = c

In general, if the limit of a function as x approaches a number from the left OR from the right decreases/increases without bound, then the line x = c is a vertical asymptote to the graph.

10

Finding Limits of Special Interest

( )1

xf x

x

1 Right limitLeft limit

What’s the domain of f? _______________

Since 1 is NOT in the domain, we are interested in what is happening to the function near x = 1…this creates two limits of special interest…the right-hand limit at x = 1 and the left-hand limit at x = 1.

Example*: Find the limits of special interest for the function,

11

Limits of Special Interest

1_____lim ( )

1__

x

xf x

x

1_____lim ( )

1__

x

xf x

x

From our limit results, we can conclude that the function has a vertical asymptote at x = 1.

12

Trig Example*

( ) cotf x xFind the vertical asymptotes of the function,

( ) cotf x x

3) Write the equation(s) of the vertical asymptotes.

4) Graph on your calculator to reinforce your result.

1) Define the function in terms of sine and cosine.

2) Determine the domain.

13

Example 1Determine the vertical asymptotes of the graph.

1) ( ) secf x x

14

Example 2Use limits to determine any vertical asymptotes of the graph.

2

22) ( )

2 15f x

x x

15

Example 3Use limits to determine any vertical asymptotes of the graph.2

2

7 123) ( )

16

x xf x

x

16

Properties of Infinite Limits

Let c and L be real numbers, let f and g be functions such that

and .

1.

2.

3.

**Similar properties would hold if

lim ( )x cf x

lim ( )

x cg x L

lim[ ( ) ( )]x c

f x g x

lim[ ( ) ( )] 0x c

f x g x if L

lim[ ( ) ( )] 0x c

f x g x if L

( )lim 0

( )x c

g x

f x

lim ( )x cf x

17

Examples of Properties of Infinite Limits

0

11) lim 5

x x

0

22) lim

cotx

x

x

3

53) lim

3x

x

x

18

Homework

Section 1.5: page 88 #1-23 odd, 29-45 odd, 49, 51