Vertical Asymptotes

Vertical asymptotes are vertical lines that a graph approaches but never touches. They occur when the function approaches a certain value as the input approaches a certain value, but the function never actually reaches that value.

How to Find Vertical Asymptotes

To find the vertical asymptotes of a function, you need to determine when the function's denominator equals zero but the numerator doesn't. This is because division by zero is undefined, so the function approaches infinity or negative infinity as it gets close to these values.

For example, if you have the function f(x) = (x + 3) / (x^2 - 4), the denominator equals zero when x = 2 or x = -2. However, the numerator doesn't equal zero at these values. Therefore, the vertical asymptotes for this function occur at x = 2 and x = -2.

Study Guide

When studying vertical asymptotes, make sure to focus on the following key points:

1. Understand the concept of vertical asymptotes and how they relate to the behavior of a function as the input approaches certain values.
2. Practice finding vertical asymptotes by identifying the values of the input that make the denominator of a function equal to zero but not the numerator.
3. Work through examples of functions with vertical asymptotes to see how they affect the overall graph of the function.
4. Review the rules for determining the behavior of a function near its vertical asymptotes (i.e., approaching positive or negative infinity).

By mastering the concept of vertical asymptotes and practicing the process of finding them, you'll be well-prepared to understand and work with this important aspect of functions and their graphs.