Question

How do you identify the horizontal asymptotes for the following function? `f(x)=(x-8)/(x+3)`. `y=`?

Answer 1

`y=1`

Explanation:

When it comes to horizontal asymptotes, there is an easy way to find them is you are given `(f(x))/(g(x))` where `f(x)` and `g(x)` are both in the form `A+Bx+Cx^2+Dx^3+cdots`

First, to determine horizontal asymptotes, look at the degree with the highest power in `f(x)` and `g(x)`.

So, in `3x^5-2x^2+3x`, `x_"max"=3x^5`

In `-2x^2+4x-1`, `x_"max"=-2x^2`

1. `f(x)_text(max)>g(x)_text(max), "no horizontal asymptotes"`

2. `f(x)_ "max">g(x)_"max", y=(f(x)_text(max))/(g(x)_text(max))`

3. `f(x)_text(max) < g(x)_text(max), y=0`

Here:
`f(x)=x-8` and `f(x)_text(max)=x`

`g(x)=x+3` and `g(x)_"max"=x`

Since both powers are the same (`x^1`) we use the 2nd rule:
`x/x=1` so the horizontal asymptote will be `y=1`