Question

What is the equation of the oblique asymptote `f(x) = (x^2-5x+6)/(x+4)`?

Answer 1

The line `y = x-9` is an asymptote on both the left and the right.

Explanation:

Rewrite the function using either Polynomial Long Division or the method below.

`f(x) = (x^2-5x+6)/(x+4)`

By long division of polynomials (or, in this case synthetic division will also work), we can get:

`f(x) = x-9 + 42/(x+4)`.

So `y = x-9` is an asymptote.

Long division is a but tedious (difficult) to format nicely here, so I'll show another way to think of this.

`x(x+4) = x^2 +4x` Use this to rewrite:

`f(x) = (x^2+4x-9x+6)/(x+4)`

Notice that we replaced the `-5x` by `4x-9x`. This is done so that we can group and reduce.

We have:

`f(x) = (x^2+4x-9x+6)/(x+4) = (x^2+4x)/(x+4) + (-9x+6)/(x+4)`

`= x + (-9x+6)/(x+4)`

Now, we'll use `-9(x+4) = -9x-36` to get:

`f(x) = x + (-9x-36+42)/(x+4)` (Replacing `+6` with `-36+42` to get:

`f(x) = x + (-9x-36)/(x+4) +42/(x+4)`

`= x-9 + 42/(x+4)`

Writing:

`f(x) = x-9 + 42/(x+4)` we can see that the difference is:

`f(x) - (x-9) = 42/(x+4)`

As `xrarroo` and as `xrarr -oo`, this difference goes to `0`, so

the line `y = x-9` is an asymptote for `f(x)` (on both sides).