Question

How do you find the vertical, horizontal and slant asymptotes of: `(x^2 - 5)/( x+3)`?

Answer 1

The vertical asymptote is `x=-3`
The slant asymptote is `y=x-3`
There is no horizontal asmptote

Explanation:

The domain of the function is `RR-(-3)`
As we cannot divide by 0

As the degree of the numerator is greater than the degree of the denominator, we would expect a slant asymptote. So we make a long division

`x^2` `color(white)(aaaa)` `-5` `∣` `x+3`
`x^2+3x` `color(white)(aa)` `∣` `x-3`
`0-3x-5`
`color(white)(aa)` `-3x-9`
`color(white)(aaaaa)` `0+4`

So we can rewrite the function
`(x^2-5)/(x+3)=x-3+4/(x+3)`

So the slant asymptote is `y=x-3`