Vertical and Horizontal Asymptotes

The line x = a is a vertical asymptote of the graph of f if f(x) approaches ∞ or f(x) approaches -∞. It corresponds with the zeroes in the denominator of a rational function.

Example: f(x) = 2x^3
                       x^2-1

To find the vertical asymptote, you should set the denominator equal to zero. The denominator can't equal zero, so the vertical asymptote is the numbers that will make the denominator equal zero

x^2-1 = 0                    Factor the equation.
(x-1)(x+1) = 0            
x = 1, -1

There are two vertical asymptotes: x = 1, -1. 

The actual graph will never pass the vertical asymptotes because they restrict the graph's behavior and where it can and can't cross. Therefore, the graph will never touch or pass the vertical asymptote.

However, the graph is allowed to touch and even pass the horizontal asymptote. The horizontal asymptote gives a general idea of the graph's behavior horizontally, but do to the fact that it can sometimes be touched or passed, it is not as specific as the vertical asymptote is about the graph's behavior.

In order to find the horizontal asymptote you must take the ratio of the leading coefficients of the denominator and numerator. Only the leading coefficients matter when determining the horizontal asymptote.

There are three situations depending on the degree of the first term of the denominator and numerator:

Situation #1
If the degree of the numerator and denominator is the same, then the horizontal asymptote is the numerator divided by the denominator. (x --> ∞, f(x) --> 2)

2x^2
x^2 -1 = 2

So in this case the horizontal asymptote is y = 2. Note that the -1 doesn't matter because the numbers with the highest degree will be the largest in comparison to the numbers that follow, making them more dominate and more important when finding the horizontal asymptote.


Situation #2
If the degree of the numerator is lower than that of the denominator, then the horizontal asymptote is y = 0. The horizontal asymptote is zero because as the denominator gets bigger, the numbers in the numerator get closer to zero (x --> ∞, f(x) --> 0), logically making the asymptote zero.

2x^2
x^3-1 = 0

So the horizontal asymptote is y = 0.

Situation #3
If the degree of the numerator is bigger than the degree of the denominator, then the horizontal asymptote is nonexistent. This is because as the numerator approaches infinity, so does the denominator (x --> ∞, f(x) --> ∞). Therefore, there is no point in a horizontal asymptote because there are no horizontal limitations to where the graph can/can't pass or touch.

2x^3
x^2-1 = none

There is no horizontal asymptote.

Helpful site:
http://www.purplemath.com/modules/asymtote.htm