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# At a Glance - Finding Vertical Asymptotes

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 1x 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 and common factors, 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 f(x).

#### Example 2

 Let . Find all vertical asymptotes (if any) of f(x).

#### Example 3

 Let Find the vertical asymptote(s) of f, if there are any.

#### Example 4

 Let f(x) = 5x2 + 3. Find the vertical asymptote(s) of f, if there are any.

#### 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.

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