Vertical Asymptotes as Infinite Limits PDF

Preview

Summary

T. Pham...

Description

Vertical Asymptotes as Infinite Limits • The concept of an infinite limit is closely related to the concept of a vertical asymptote. Consider the graph of the function f(x) equals 2 to the power of 1 over quantity x minus 1. You can see that as the graph approaches x equals 1 from the right side, the graph rises vertically without bound.

• So, the graph has a vertical asymptote at x equals 1. • You can use a 1-sided limit to describe the behavior of the graph to the right of 1. • The limit of the function as x approaches 1 from the right side is equal to infinity Vertical Asymptote Definition • If f(x) approaches infinity (or negative infinity) as x approaches c from the right or from the left, then the line x=c is a vertical asymptote of the graph of f. Infinite Limit Definition • A limit in which a function f(x) approaches infinity (or negative infinity) as x approaches from the right or from the left is called an infinite limit

Vertical Asymptote of a Rational Function • The graph of the rational function: • • Has a vertical asymptote at x=c if q(c)=0 and p(c)=0 • To find the vertical asymptotes of a rational function, simply find the zeros of the denominator and verify that they are not also zeros of the numerator

• There is a hole in the graph because x equals 0 is a zero of both the denominator and the numerator

• The denominator has zeros at x=0, x=1, and x=3. But x=1 is also a zero of the numerator. So f has 2 vertical asymptotes, x=0, x=3 • To find the 1-sided Limits, use a graphic calculator to graph f

• From the graph, you can conclude the following...