Step by step
The question
Question: find the asymptotes and holes of f(x) = 3x/(x² − 4).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 Factor the bottom
The bottom: x² − 4.
x² − 4 is x² minus 2 squared.
A square minus a square is the plus bracket times the minus bracket.
So the bottom is (x + 2)(x − 2).
2 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
The bottom: x + 2 = 0 when x = −2.
Look: at x = −1.9, the bottom is tiny.
So f(x) is about 15: huge.
And x − 2 = 0 when x = 2.
3 Horizontal asymptote
For a very big x, the biggest power on the top and on the bottom is all that counts.
The rest is tiny next to it.
The top has a smaller power than the bottom.
When x is huge, the bottom is much bigger, so f(x) gets close to 0.
Look: at x = 100, f(x) is about 0.03.
Close to 0.
So the horizontal asymptote is y = 0.
So: x = −2, x = 2 and y = 0.
4 Check
Let's check: put x = −2 in the question's bottom.
It gives 0.
Dividing by 0 is not allowed: an asymptote.
And a very big x, 1000, gives almost exactly 0.
It works!
Let's look back.
First: factor the bottom.
Then: vertical asymptotes.
Then: horizontal asymptote.
Last: check.