Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)=x/(x^2+4)`?

Answer 1

This function has a horizontal asymptote at `y=0` and has no vertical or oblique asymptotes.

Explanation:

`f(x)=x/(x^2+4)`

To find the vertical asymptotes, solve for the values of x that make the denominator equal zero.

`x^2+4=0`
`x^2=-4`

Because there are no real values of x that will make the denominator equal zero, this function has no vertical asymptotes.

To find the horizontal asymptote, compare the degree of the numerator and denominator. The degree of the numerator is `color(limegreen)1`. The degree of the denominator is `color(red)2`.

`f(x)=x^color(limegreen)1/(x^color(red)2+4)`

If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is "pulled down" to `y=0`.

If there is a horizontal asymptote, there is no oblique asymptote.