Question

How do you find the horizontal asymptote for `f(x) = (3x) / (x+4)`?

Answer 1

I found `y=3`

Explanation:

The horizontal asymptote is a line towards which the curve, described by your function, tends to get as near as possible.

To find it you can try to see what happens to your function when `x` becomes VERY big....and see if your functions "tends" to some kind of fixed value:
as `x` becomes very big, say `x=1,000,000` you have:
`f(1,000,000)=(3*1,000,000)/(1,000,000+4)`
let us forget the `4` that is negligible compared to `1,000,000`; you have:

`f(1,000,000)=(3*cancel(1,000,000))/(cancel(1,000,000))=3`

So when `x` becomes very big positively (and negatively, you can try this) your functions "tends" to get near the value `3`!
So the horizontal line of equation `y=3` will be your asymptote!

You can plot your function and see this tendency!
graph{(3x)/(x+4) [-41.1, 41.07, -20.56, 20.53]}