1.5 Infinite Limits and 3.5 Limits at Infinity AP Calculus I Ms. Hernandez (print in grayscale or black/white)

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

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 81  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 82  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 83  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 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

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 83  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 84  Sum/difference  Product L>0, L<0  Quotient (#/infinity = 0)  Same properties for  Ex 5 pg 84

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

3.5 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

Some examples  Ex 2-3 on pages #  What’s the graph look like on Ex 3.c  Called oblique asymptotes (not in cal 1)  KNOW Guidelines on page 195

2 horizontal asymptotes  Ex 4 pg 196  Is the method for solving lim of f(x) with 2 horizontal asymptotes any different than if the f(x) only had 1 horizontal asymptotes?

Trig f(x)  Ex 5 pg 197  What is the difference in the behaviors of the two trig f(x) in this example?  Oscillating toward no value vs oscillating toward a value

Word Problems !!!!!  Taking information from a word problem and apply properties of limits at infinity to solve  Ex 6 pg 197

A word on infinite limits at infinity  Take a lim of f(x)  infinity and sometimes the answer is infinity  Ex 7 on page 198  Uses property of f(x)  Ex 8 on page 198  Uses LONG division of polynomials-Yuck!