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

Dec 20, 2015

I found:
$\textcolor{red}{\text{Vertical Asymptote: } x = 3}$
$\textcolor{b l u e}{\text{Horizontal Asymptote: } y = 1}$

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).

$\textcolor{red}{\text{Vertical Asymptote: } x = 3}$

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:

$f \left(1 , 000 , 000\right) = \frac{1 , 000 , 000}{1 , 000 , 000 - 3} \approx 1$

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

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

$\textcolor{b l u e}{\text{Horizontal Asymptote: } y = 1}$

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