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

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

Infinite Limits  If function values keep INCREASING ________________ as x approaches a given value we say the limit is _____________. WITHOUT BOUND INFINITY

Infinite Limits  If function values keep DECREASING ________________ as x approaches a given value we say the limit is _____________. WITHOUT BOUND - INFINITY

IMPORTANT NOTE: The equal sign in the statement does NOT mean the limit exists! On the contrary, it tells HOW the limit FAILS to exist.

Examples

REMEMBER: The equal sign in the statement does NOT mean the limit exists! On the contrary, it tells HOW 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.

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)

Determining Infinite Limits from a Graph  Example 1 pg 84  Can you get different infinite limits from the left or right of a graph?  How do you find the vertical asymptote?

Finding Vertical Asymptotes  Ex 2 pg 84  Denominator = 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.

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 right

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

Determining Infinite Limits  Ex 4 pg 86  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!

Properties of Infinite Limits  Page 87  Sum/difference  Product L>0, L<0  Quotient (#/infinity = 0)  Same properties for  Ex 5 pg 87

Asymptotes & Limits at Infinity For the function, find (a) (b) (c) (d) (e) All horizontal asymptotes (f) All vertical asymptotes

Asymptotes & Limits at Infinity For x>0, |x|=x (or my x-values are positive) 1/big = little and 1/little = big sign of denominator leads answer For x<0 |x|=-x (or my x-values are negative) 2 and –2 are HORIZONTAL Asymptotes

Asymptotes & Limits at Infinity

1.5 Limit at Infinity  Horizontal asymptotes!  Lim as x  infinity of f(x) = horizontal asymptote  #/infinity = 0  Infinity/infinity  Divide the numerator & denominator by a denominator degree of x