- Topics At a Glance
- Limits
- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
**All About Asymptotes**- Vertical Asymptotes
**Finding Vertical Asymptotes**- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
- Manipulating Limits
- The Basic Properties
- Multiplication by a Constant
- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.