Infinite Limits

Lesson 1.5

Infinite Limits

Two Types of infinite limits.

Either the limit equals infinity or the limit is approaching infinity.

We are going to take a look at when the limit equals infinity, for now.

1.5 Infinite Limits

• Vertical asymptotes at x = c will give you infinite limits

• Take the limit at x = c and the behavior of the graph at x = c is a vertical asymptote then the limit is infinity

• Really the limit does not exist, and that it fails to exist is b/c of the unbounded behavior (and we call it infinity)

The function f(x) will have a vertical asymptote at x = a if we obtain any of

the following limits:

)(lim xfax

)(lim xfax

)(lim xfax

Definition of Infinite Limits

M --------------

f(x) increases without bound as x c

NOTE: may decrease without bound ie: go to negative infinity!!

Vertical Asymptotes

• When f(x) approachesinfinity as x → c– Note calculator often

draws false asymptote

• Vertical asymptotes generated byrational functions when g (x) = 0

c

( )( )

( )

f xh x

g x

Theorem 1.14Finding Vertical Asymptotes

• If the denominator = 0 at x = c AND the numerator is NOT zero, we have a vertical asymptote at x = c!!!!!!! IMPORTANT

• What happens when both num and den are BOTH Zero?!?!

A Rational Function with Common Factors(Should be x approaching 2)

• When both numerator and denominator are both zero then we get an indeterminate form and we have to do something else …

– Direct sub yields 0/0 or indeterminate form– We simplify to find vertical asymptotes but how do we

solve the limit? When we simplify we still have indeterminate form.

2

22

2 8lim

4x

x x

x

2

4lim , 2

2x

xx

x

A Rational Function with Common Factors, cont….

• Direct sub yields 0/0 or indeterminate form. When we simplify eliminate indeterminate form and we learn that there is a vertical asymptote at x = -2 by theorem 1.14.

• Take lim as x-2 from left and right

• Take values close to –2 from the right and values close to –2 from the left … Table and you will see values go to positive or negative infinity

2

22

2 8lim

4x

x x

x

2

22

2 8lim

4x

x x

x

Determining Infinite Limits

• Denominator = 0 when x = 1 AND the numerator is NOT zero– Thus, we have vertical

asymptote at x=1

• But is the limit +infinity or –infinity?

• Let x = small values close to c

• Use your calculator to make sure – but they are not always your best friend!

2 2

1 1

3 3 lim and lim

1 1x x

x x x xFind

x x

Infinite Limits:

1f x

x

0

1limx x

As the denominator approaches zero, the value of the fraction gets very large.

If the denominator is positive then the fraction is positive.

0

1limx x

If the denominator is negative then the fraction is negative.

vertical asymptote at x=0.

Example 4:

20

1limx x

20

1limx x

The denominator is positive in both cases, so the limit is the same.

20

1 limx x

Properties of Infinite Limits• Given

Then• Sum/Difference

• Product

• Quotient

lim ( ) and lim ( )x c x cf x g x L

lim ( ) ( )x c

f x g x

lim ( ) ( ) 0x c

f x g x L

( )lim 0

( )x c

g x

f x

lim ( ) ( ) 0x c

f x g x L

Find each limit, if it exists.

4

11. lim

4x x

6

4

2

-2

-4

-6

-5 5

Find each limit, if it exists.

4

11. lim

4x x

1

3.999 4

1

VS

Very small negative #

One-sided limits will always exist!

6

4

2

-2

-4

-6

-5 5

1

12. lim

1x x

6

4

2

-2

-4

-6

-5 5

1

12. lim

1x x

1

0.999 1

1

VS

This time we only care if the two sides come together—and where.

6

4

2

-2

-4

-6

-5 5

1

1lim

1x x

1

1lim

1x x

1

1.001 1

1

VS

DNE

Can’t do Direct Sub, need to go to our LAST resort…

check the limits from each side.

3. Find any vertical asymptotes of2

2

2 8( )

4

x xf x

x

6

4

2

-2

-4

-6

-5 5

3. Find any vertical asymptotes of2

2

2 8( )

4

x xf x

x

Discontinuous at x = 2 and -2.

4 2

2 2

x x

x x

4

2

x

x

V.A. at x = -2 3

2Hole at 2,

6

4

2

-2

-4

-6

-5 5

Try It Out

• Find vertical asymptote

• Find the limit

• Determine the one sided limit

2

2( )

1

xg x

x x

2

24lim

16x

x

x

3

2 1

1( ) lim ( )

1 x

xf x f x

x x

Methods

• Visually: Graphing• Analytically: Make a table close to “a”• Substitution: Substitute “a” for x

If Substitution leads to:1) A number L, then L is

the limit

2) 0/k, then the limit is

zero

3) k/0, then the limit is ±∞, or

dne

4) 0/0, an indeterminant form, you must do more!