- 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

If *f*(*x*) is a function and there's some number *L* with

we draw a dashed horizontal line on the graph at height *y* = *L*. This line is called a **horizontal asymptote**. As *x* approaches ∞, the function *f*(*x*) gets closer to *L*, in the graph the function gets closer to the dashed horizontal line.

We also draw a horizontal asymptote at *y* = *L* if

Then as *x* approaches -∞, or as we move left on the graph, the function *f(x)* will approach the dashed horizontal line.

It is fine for a graph to cross over its horizontal asymptote(s). We can have something like this, for example:

The important thing is that as *x* gets bigger (or more negative), the function is getting closer to the horizontal asymptote.

Finding the horizontal asymptote(s) of a function is the same task as finding the limits of a function *f(x)* as *x* approaches ∞ or -∞.

The difference is that horizontal asymptotes are drawn as dashed horizontal lines in a graph, while limits (when they exist) are numbers.