infinite limits lesson 1.5. infinite limits two types of infinite limits. either the limit equals...

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!