# How do you find the asymptotes for y = (7x-5)/(2-5x)?

##### 1 Answer

The asymptotes are $x = \frac{2}{5}$ vertical asymptote
$y = - \frac{7}{5}$ horizontal asymptote

#### Explanation:

Take the limit of y as x approaches $\infty$

${\lim}_{x \to \infty} y = {\lim}_{x \to \infty} \frac{7 x - 5}{- 5 x + 2} = {\lim}_{x \to \infty} \frac{7 - \frac{5}{x}}{- 5 + \frac{2}{x}} = - \frac{7}{5}$

$x = - \frac{7}{5}$

Also if you solve for x in terms of y,

$y = \frac{7 x - 5}{- 5 x + 2}$

$y \left(- 5 x + 2\right) = 7 x - 5$

$- 5 x y + 2 y = 7 x - 5$

$2 y + 5 = 7 x + 5 x y$

$2 y + 5 = x \left(7 + 5 y\right)$

$x = \frac{2 y + 5}{5 y + 7}$

take now the limit of x as y approaches $\infty$

${\lim}_{y \to \infty} x = {\lim}_{y \to \infty} \frac{2 y + 5}{5 y + 7} = {\lim}_{y \to \infty} \frac{2 + \frac{5}{y}}{5 + \frac{7}{y}} = \frac{2}{5}$

$y = \frac{2}{5}$

kindly see the graph.

graph{y=(7x-5)/(-5x+2)[-20,20,-10,10]}

have a nice day!