- 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

Both vertical asymptotes and holes are places where the curve never quite touches. Holes occur at places where the limit of the function exists but the function itself does not. These correspond to the roots (or zeros) of the denominator that cancel out entirely during simplification. Vertical asymptotes occur at places where the limit of the function is ∞ or -∞, which happen at the roots of the denominator that are left over after simplification.

Example 1

Find all vertical asymptotes and/or holes of the function |

Example 2

Find all vertical asymptotes and/or holes of the function |

Exercise 1

Find all vertical asymptotes and holes for the rational function below. Give both the *x* and *y* coordinates for the holes.

Exercise 2

Find all vertical asymptotes and holes for the rational function below. Give both the *x* and *y* coordinates for the holes.

Exercise 3

Find all vertical asymptotes and holes for the rational function below. Give both the *x* and *y* coordinates for the holes.

Exercise 4

*x* and *y* coordinates for the holes.

Exercise 5

*x* and *y* coordinates for the holes.