How do I find the vertical and horizontal asymptotes of the function f(x)=(3x-1)/(x+4)?

Apr 3, 2018

See below.

Explanation:

$f \left(x\right) = \frac{3 x - 1}{x + 4}$

Vertical asymptotes occur where the function is undefined.

This happens when $x = - 4$

So the line $\textcolor{b l u e}{x = - 4}$ is a vertical asymptote.

We next examine the end behaviour as $x \to - \pm \infty$

Starting with:

$\frac{3 x - 1}{x + 4}$

Divide by $x$:

$\frac{\frac{3 x}{x} - \frac{1}{x}}{\frac{x}{x} + \frac{4}{x}}$

Cancelling:

$\frac{\frac{3 \cancel{x}}{\cancel{x}} - \frac{1}{x}}{\cancel{\frac{x}{x}} + \frac{4}{x}} = \frac{3 - \frac{1}{x}}{1 + \frac{4}{x}}$

as $x \to \infty$ , $\frac{3 - \frac{1}{x}}{1 + \frac{4}{x}} \to \frac{3 - 0}{1 + 0} = 3$

as $x \to - \infty$ , $\frac{3 - \frac{1}{x}}{1 + \frac{4}{x}} \to \frac{3 - 0}{1 + 0} = 3$

This show that the line $\textcolor{b l u e}{y = 3}$ is a horizontal asymptote.

These findings are confirmed by the graph of $f \left(x\right) = \frac{3 x - 1}{x + 4}$: 