Question

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

Answer 1

I found `y=4`

Explanation:

The horizontal asymptote is a horizontal line of equation: `y="constant"`
towards which the curve described by your function TENDS to get closer and closer maybe not immediately but as `x` becomes sufficently big.

To find this line there is a trick!

Take your function and try to "see" its behavior very far from the origin...i.e. when `x` becomes VEEEEERY big!
In your case consider a `x` value very big, say, `x=1,000,000`:

you get:
`y=4*(1,000,000)/(1,000,000-3)~~4*(1,000,000)/(1,000,000)=` the `3` is negligible;
`y=4*(cancel(1,000,000))/(cancel(1,000,000))`
So, you get `y=4` that is the equation of a horizontal line that your function tends to become for `x` VEEEERY large!!
Your asymptote!!!

You can "see" this graphically:
graph{(4x)/(x-3) [-25.66, 25.65, -12.83, 12.83]}

The two branches of your function will get as near as possible to the horizontal line `y=4`!

Hope it helps!