Question

How to find the asymptotes of `f(x) = (3x^2 + 2x - 1 )/( x + 1)`?

Answer 1

No asymptotes.

Explanation:

`f(x)=(3x^2+2x-1)/(x+1)`

Factoring the numerator:

`((3x-1)(x+1))/(x+1)`

`((3x-1)cancel((x+1)))/cancel((x+1))`

This is known as a removable discontinuity:

`f(x)=3x-1`

So we are left with a linear function. These have no asymptotes.

NOTE

It is best to check rational functions for removable discontinuities first, otherwise you can end up making errors. If we had not noticed the discontinuity in the above function, we would have calculated there to be an asymptote at `x=-1`. This is where the function would have been undefined, but as we have shown there are no asymptotes because we have a linear function.

Graph:

graph{(3x^2+2x-1)/(x+1) [-10, 10, -5, 5]}