# How do you find the asymptotes for f(x) = 5/(x - 7) + 6?

Jun 6, 2015

Given:
$y = f \left(x\right) = \frac{5}{x - 7} + 6$

Subtract $6$ from both sides to get:

$y - 6 = \frac{5}{x - 7}$

Multiply both sides by $\left(x - 7\right)$ and divide both sides by $\left(y - 6\right)$ to get:

$x - 7 = \frac{5}{y - 6}$

Add $7$ to both sides to get:

$x = \frac{5}{y - 6} + 7$

The asymptotes correspond to the excluded values:

$x = 7$ and $y = 6$

Jun 6, 2015

The one asymptote is when the numerator of the fraction gets closer to $0$. This is when $x \to 7$, so $x = 7$ is the vertical asymptote.

The other one is when $x$ goes very large. The fraction will become smaller, so the function as a whole will get nearer to $6$ without reaching it. So $y = 6$ is the horizontal asymptote.
graph{5/(x-7) +6 [-18.87, 46.08, -9.32, 23.12]}