# Asymptotes

## Key Questions

• To Find Vertical Asymptotes:

In order to find the vertical asymptotes of a rational function, you need to have the function in factored form. You also will need to find the zeros of the function. For example, the factored function $y = \frac{x + 2}{\left(x + 3\right) \left(x - 4\right)}$ has zeros at x = - 2, x = - 3 and x = 4.

*If the numerator and denominator have no common zeros, then the graph has a vertical asymptote at each zero of the denominator. In the example above $y = \frac{x + 2}{\left(x + 3\right) \left(x - 4\right)}$, the numerator and denominator do not have common zeros so the graph has vertical asymptotes at x = - 3 and x = 4.

*If the numerator and denominator have a common zero, then there is a hole in the graph or a vertical asymptote at that common zero.
Examples:
1. $y = \frac{\left(x + 2\right) \left(x - 4\right)}{x + 2}$ is the same graph as y = x - 4, except it has a hole at x = - 2.

2.$y = \frac{\left(x + 2\right) \left(x - 4\right)}{\left(x + 2\right) \left(x + 2\right) \left(x - 4\right)}$ is the same as the graph of $y = \frac{1}{x + 2} ,$ except it has a hole at x = 4. The vertical asymptote is x = - 2.

To Find Horizontal Asymptotes:

• The graph has a horizontal asymptote at y = 0 if the degree of the denominator is greater than the degree of the numerator. Example: In $y = \frac{x + 1}{{x}^{2} - x - 12}$ (also $y = \frac{x + 1}{\left(x + 3\right) \left(x - 4\right)}$ ) the numerator has a degree of 1, denominator has a degree of 2. Since the degree of the denominator is greater, the horizontal asymptote is at $y = 0$.

• If the degree of the numerator and the denominator are equal, then the graph has a horizontal asymptote at $y = \frac{a}{b}$, where a is the coefficient of the term of highest degree in the numerator and b is the coefficient of the term of highest degree in the denominator. Example: In $y = \frac{3 x + 3}{x - 2}$ the degree of both numerator and denominator are both 1, a = 3 and b = 1 and therefore the horizontal asymptote is $y = \frac{3}{1}$ which is $y = 3$

• If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote.

• Example 1: $f \left(x\right) = {x}^{2} / \left\{\left(x + 2\right) \left(x - 3\right)\right\}$

Vertical Asymptotes: $x = - 2$ and $x = 3$
Horizontal Asymptote: $y = 1$
Slant Asymptote: None

Example 2: $g \left(x\right) = {e}^{x}$

Vertical Asymptote: None
Horizontal Asymptote: $y = 0$
Slant Asymptote: None

Example 3: $h \left(x\right) = x + \frac{1}{x}$

Vertical Asymptote: $x = 0$
Horizontal Asymptote: None
Slant Asymptote: $y = x$

I hope that this was helpful.

• An asymptote is a value of a function that you can get very near to, but you can never reach.

Let's take the function $y = \frac{1}{x}$
graph{1/x [-10, 10, -5, 5]}
You will see, that the larger we make $x$ the closer $y$ will be to $0$
but it will never be $0$ $\left(x \to \infty\right)$

In this case we call the line $y = 0$ (the x-axis) an asymptote

On the other hand, $x$ cannot be $0$ (you can't divide by$0$)

So the line $x = 0$ (the y-axis) is another asymptote.