# What are the vertical and horizontal asymptotes of f(x)=(3x-1)/(x+4)?

Mar 26, 2018

Horizontal Asymptote: $y = 3$
Vertical Asymptote : $x = - 4$

#### Explanation:

Since we have to find a vertical Asymptote, we must find a x value that makes the function of form anything/0 since it reaches to infinity. See in the denominator $x = - 4$ will do something like anything over zero. So $x = - 4$ is a vertical Asymptote.

For horizontal Asymptote. We must figure out what happens to the function when it reaches infinity. Since we can't figure what happens to the function right in this form. So we must do some tricks. Recall $\frac{1}{\infty} \rightarrow 0$. It works for every number
Given $f \left(x\right) = \frac{3 x - 1}{x + 4}$
Divide numerator and denominator by x .
$= \frac{\left(\frac{3}{x}\right) - \left(\frac{1}{x}\right)}{\left(\frac{x}{x}\right) + \left(\frac{4}{x}\right)}$
$= \frac{3 - \frac{1}{x}}{1 + \frac{4}{x}}$
When $x \rightarrow \infty$ the number with denominator will shrink down to zero
$= \frac{3 - 0}{1 + 0}$
$= \frac{3}{1}$
Hence $y = 3$
Is the horizontal Asymptote.