# Functions, Graphs, and Limits

### Topics

## Introduction to Functions, Graphs, And Limits - At A Glance:

Vertical asymptotes most frequently show up in rational functions. When a rational function *f*(*x*) has a non-zero constant in the numerator and an expression with a variable in the denominator, the function *f*(*x*) will have vertical asymptotes at all values of *x* that make the denominator 0. If the denominator has no roots, then *f*(*x*) will have no vertical asymptotes.

This works for the same reason that 1 / x has a vertical asymptote at zero: the numerator is a non-zero constant and the denominator is getting smaller and smaller, therefore the fraction will get bigger and bigger.

If we use a different constant, the principle is still the same.

If a rational function has something besides a non-zero constant in the numerator, we may need to be creative: factor the numerator and denominator, cancel if possible, and find the roots of whatever polynomial remains in the denominator after factoring.

Be Careful: Remember to simplify the rational functions!

#### Example 1

Let . Find all vertical asymptotes (if any) of |

#### Example 2

Let . Find all vertical asymptotes (if any) of |

#### Example 3

Let Find the vertical asymptote(s) of |

#### Example 4

Let |

#### Example 5

Find the vertical asymptotes of |

#### Exercise 1

Find all vertical asymptotes, if any, for the function.

#### Exercise 2

Find all vertical asymptotes, if any, for the function.

#### Exercise 3

Find all vertical asymptotes, if any, for the function.

#### Exercise 4

Find all vertical asymptotes, if any, for the function.

#### Exercise 5

Find all vertical asymptotes, if any, for the function.

#### Exercise 6

- What function is pretending to be near
*x*= 7?

#### Exercise 7

- What function is pretending to be near
*x*= 4?

#### Exercise 8

- What function is pretending to be near
*x*= -9?

#### Exercise 9

Find the vertical asymptote(s) (if any) for the function.

#### Exercise 10

Find the vertical asymptote(s) (if any) for the function.

#### Exercise 11

Find the vertical asymptote(s) (if any) for the function.

#### Exercise 12

Find the vertical asymptote(s) (if any) for the function.

#### Exercise 13

Find the vertical asymptote(s) (if any) for the function.