1.5 Infinite Limits

IB/AP Calculus I

Ms. Hernandez

Modified by Dr. Finney

AP Prep Questions / Warm Up

No Calculator!

(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent

(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE

1

lnlimx

x

x

22

( 2)lim

4x

x

x

AP Prep Questions / Warm Up

No Calculator!

(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent

(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE

1

ln ln1 0lim 0

1 1x

x

x

22 2 2

( 2) ( 2) 1 1lim lim lim

4 ( 2)( 2) ( 2) 4x x x

x x

x x x x

Infinite Limits If function values keep INCREASING

________________

as x approaches a given value

we say the limit is _____________.

WITHOUT BOUNDWITHOUT BOUND

INFINITY

lim ( )x cf x

Infinite Limits If function values keep DECREASING

________________

as x approaches a given value

we say the limit is _____________.

WITHOUT BOUNDWITHOUT BOUND

- INFINITY

lim ( )x cf x

IMPORTANT NOTE:

The equal sign in the statement

does NOT mean the

limit exists!

lim ( )x cf x

On the contrary, it tells HOW On the contrary, it tells HOW the limit FAILS to exist.the limit FAILS to exist.

Examples

2

1lim

2x x

2

1: lim

2x

THINKx

2

1: lim

2x

THINKx

Examples

22

1: lim

2x

THINKx

22

1: lim

2x

THINKx

22

1lim

2x x

REMEMBER:

The equal sign in the statement

does NOT mean the

limit exists!

lim ( )x cf x

On the contrary, it tells HOW On the contrary, it tells HOW the limit FAILS to exist.the limit FAILS to exist.

Definition of a Vertical Asymptote

If f(x) approaches infinity or negative infinity as x approaches c from the left or right,

then x = c is a vertical asymptote of f.

@ lim ( ) lim ( )x c x c

VA x c if f x or f x

1.5 Infinite LimitsVertical asymptotes at x=c will give you

infinite limitsTake 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)

Determining Infinite Limits from a Graph

Example 1 pg 84Can you get different infinite limits from

the left or right of a graph?How do you find the vertical asymptote?

Finding Vertical AsymptotesEx 2 pg 84Denominator = 0 at x = c AND the

numerator is NOT zero Thus, we have vertical asymptote at x = c

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

A Rational Function with Common Factors When both num and den are both zero then

we get an indeterminate form and we have to do something else …

Ex 3 pg 86

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

Ex 3 pg 86: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2.

Take lim as x-2 from left and right2

22

2 8lim

4x

x x

x

2

22

2 8lim

4x

x x

x

A Rational Function with Common Factors Ex 3 pg 83: Direct sub yields 0/0 or indeterminate form.

When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2.

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 LimitsEx 4 pg 86Denominator = 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 cUse your calculator to make sure – but

they are not always your best friend!

Properties of Infinite LimitsPage 87

Sum/differenceProduct L>0, L<0Quotient (#/infinity = 0)Same properties for Ex 5 pg 87

lim ( )x cf x

lim ( )x cg x L

lim ( )x cf x

Asymptotes & Limits at InfinityFor the function , find(a)

(b)

(c)

(d)

(e) All horizontal asymptotes(f) All vertical asymptotes

2 1( )

xf x

x

lim ( )x

f x

lim ( )x

f x

0lim ( )x

f x

0lim ( )x

f x

Asymptotes & Limits at Infinity

For x>0, |x|=x (or my x-values are positive)

1/big = little and 1/little = bigsign of denominator leads answerFor x<0 |x|=-x (or my x-values are negative)

2 and –2 are HORIZONTAL Asymptotes

2 1( )

xf x

x

2 1 2 1 1lim ( ) lim lim lim 2 2x x x x

x xf x

x x x

2 1 2 1 1lim ( ) lim lim lim 2 2x x x x

x xf x

x x x

Asymptotes & Limits at Infinity2 1

( )x

f xx

0 0 0 0

2 1 2 1 1lim ( ) lim lim lim 2x x x x

x xf x

x x x

2 1 2 1 1lim ( ) lim lim lim 2 2x x x x

x xf x

x x x

1 12 2 2 limDNEx little

1 12 2 2 limDNEx little

1.5 Limit at InfinityHorizontal asymptotes!Lim as xinfinity of f(x) = horizontal

asymptote#/infinity = 0 Infinity/infinity

Divide the numerator & denominator by a denominator degree of x