Question

How do you find vertical, horizontal and oblique asymptotes for `(3x+5)/ (x-6)`?

Answer 1

The vertical asymptote is `x=6`
The horizontal asymptote is `y=3`
There are no oblique asymptote

Explanation:

The function is not defined when `x=6` as we cannot divide by zero
So we have a vertical asymptote at `x=6`

As the degree of the numerator is identical to the degree of the denominator, so we make a long division

`3x+5` `color(white)(aaaa)` `∣` `x-6`
`3x-18` `color(white)(aaa)` `∣` `3`
`0-23`

Finally we obtain
`(3x+5)/(x-6)=3+23/(x-6)`
So `y=3` is a horizontal asymptote

We could get the same result by finding the limit as `x->+-oo`
limit `(3x+5)/(x-6)=3`
`x->+-oo`