Step by step
The question
Question: find the asymptotes and holes of f(x) = (2x² − 8)/(x² − 1).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 Factor the top and bottom
The top: 2x² − 8.
Every part can be divided by 2.
We take it out in front.
2x² − 8 is x² minus 2 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top is 2(x + 2)(x − 2).
The bottom: x² − 1.
x² − 1 is x² minus 1 squared.
A square minus a square is the plus bracket times the minus bracket.
So the bottom is (x + 1)(x − 1).
2 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
The bottom: x + 1 = 0 when x = −1.
Look: at x = −0.9, the bottom is tiny.
So f(x) is about 34: huge.
And x − 1 = 0 when x = 1.
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 same power on top and bottom.
The x's cancel, and the numbers in front give 2.
Look: at x = 100, f(x) is about 2.
Close to 2.
So the horizontal asymptote is y = 2.
So: x = −1, x = 1 and y = 2.
4 Check
Let's check: put x = −1 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 2.
It works!
Let's look back.
First: factor the top and bottom.
Then: vertical asymptotes.
Then: horizontal asymptote.
Last: check.