How do you find the vertical, horizontal or slant asymptotes for f(x) = (2x-8) / (3x-24)?

Jul 20, 2016

vertical asymptote x = 8
horizontal asymptote $y = \frac{2}{3}$

Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: 3x - 24 = 0 → x = 8 is the asymptote

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide terms on numerator/denominator by x

$\frac{\frac{2 x}{x} - \frac{8}{x}}{\frac{3 x}{x} - \frac{24}{x}} = \frac{2 - \frac{8}{x}}{3 - \frac{24}{x}}$

as $x \to \pm \infty , f \left(x\right) \to \frac{2 - 0}{3 - 0}$

$\Rightarrow y = \frac{2}{3} \text{ is the asymptote}$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case her ( both degree 1 ) Hence there are no slant asymptotes.
graph{(2x-8)/(3x-24) [-20, 20, -10, 10]}