Question

How to find the asymptotes of `y = x /(x-3)` ?

Answer 1

I found:
`color(red)("Vertical Asymptote: "x=3)`
`color(blue)("Horizontal Asymptote: "y=1)`

Explanation:

Looking at your function you can "see" a vertical asymptote,i.e., a vertical line towards which the graph of your function will tend to get near as much as possible.

You need to look for possible prohibited `x` values and there you'll find your asymptote.

In this case the prohibited `x` value is the one that makes your function a division by ZERO: i.e., `x=3` (you cannot use `x=3` into your function so the vertical line passing through it will be a forbidden place or your asymptote).

`color(red)("Vertical Asymptote: "x=3)`

You can also "see" a horizontal asymptote,i.e., a horizontal line towards which the graph of your function will tend to.

This is a little more difficult but you can think big and imagine what happens to the graph of your function when `x` becomes VERY large and ask yourself: "it tends towards something?".

Let us take a `x` very big, say, `x=1,000,000`
and evaluate your function there so you get:

`f(1,000,000)=(1,000,000)/(1,000,000-3)~~1`

because `3`, compared to `1,000,000`, is negligible.

So you get that the horizontal line passing through: `y=f(1,000,000)=1`
will be the line towards which the graph tends to get near as much as possible:

`color(blue)("Horizontal Asymptote: "y=1)`

You can see graphically these two tendencies:
graph{x/(x-3) [-8.89, 8.89, -4.444, 4.445]}