# How do I find the asymptotes of y=7/(3x-4)-1/9?

Aug 6, 2016

vertical asymptote $x = \frac{4}{3}$
horizontal asymptote $y = - \frac{1}{9}$

#### Explanation:

The denominator of y 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 - 4 = 0 $\Rightarrow x = \frac{4}{3} \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$\frac{\frac{7}{x}}{\frac{3 x}{x} - \frac{4}{x}} - \frac{1}{9} = \frac{\frac{7}{x}}{3 - \frac{4}{x}} - \frac{1}{9}$

as $x \to \pm \infty , y \to \frac{0}{3 - 0} - \frac{1}{9}$

$\Rightarrow y = - \frac{1}{9} \text{ is the asymptote}$
graph{(7)/(3x-4)-1/9 [-10, 10, -5, 5]}