Question

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

Answer 1

Given:
`y = f(x)=5/(x-7)+6`

Subtract `6` from both sides to get:

`y-6 = 5/(x-7)`

Multiply both sides by `(x-7)` and divide both sides by `(y - 6)` to get:

`x-7 = 5/(y-6)`

Add `7` to both sides to get:

`x = 5/(y-6)+7`

The asymptotes correspond to the excluded values:

`x=7` and `y=6`

Answer 2

The one asymptote is when the numerator of the fraction gets closer to `0`. This is when `x->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]}