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:

Definition of Infinite Limits f(x) increases without bound as x  c NOTE: may decrease without bound ie: go to negative infinity!! M --------------

Vertical Asymptotes When f(x) approaches infinity as x → c Note calculator often draws false asymptote Vertical asymptotes generated by rational functions when g (x) = 0 c

Theorem 1.14 Finding 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.

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

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!

Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

The denominator is positive in both cases, so the limit is the same. Example 4: The denominator is positive in both cases, so the limit is the same.

Properties of Infinite Limits Given Then Sum/Difference Product Quotient

Find each limit, if it exists.

Find each limit, if it exists. One-sided limits will always exist! Very small negative #

This time we only care if the two sides come together—and where. Can’t do Direct Sub, need to go to our LAST resort… check the limits from each side.

3. Find any vertical asymptotes of

3. Find any vertical asymptotes of Discontinuous at x = 2 and -2. V.A. at x = -2

Try It Out Find vertical asymptote Find the limit Determine the one sided limit

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 4) 0/0, an indeterminant form, you must do more! 3) k/0, then the limit is ±∞, or dne