Step by step
The question
Question: find the asymptotes of f(x) = (x² − 2x − 2)/(x − 3).
An asymptote is a line the graph gets closer and closer to.
We draw it dashed.
1 Compare the powers
The top has x²: power 2.
The bottom has x: power 1.
The top power is one bigger.
Then there is no flat asymptote.
Instead, the graph gets close to a slanted line: a slant asymptote.
Dividing finds it.
2 Set it up
We write it like long division with numbers.
x − 3 goes outside, x² − 2x − 2 inside.
3 First part of the answer
Divide the first parts: x² divided by x is x.
We write it on top.
Multiply: x times (x − 3) is x² − 3x.
We write it under, with a minus: we take it away.
x² take away x² is 0.
That is why we chose x.
−2x take away −3x is x.
Bring down the next part: −2.
4 Second part of the answer
Divide the first parts: x divided by x is 1.
We write it on top.
Multiply: 1 times (x − 3) is x − 3.
We write it under, with a minus: we take it away.
x take away x is 0.
That is why we chose 1.
−2 take away −3 is 1.
1 has no x.
We can't divide it by x.
So 1 is the remainder: what is left over.
So f(x) is x + 1, plus 1/(x − 3).
5 Read the slant asymptote
When x is big, 1/(x − 3) is tiny.
At x = 100 it is 1/97: about 0.01.
Almost 0.
So for a big x, f(x) is close to x + 1.
The slant asymptote is y = x + 1.
Careful!
The remainder part is not in the line.
It gets smaller and smaller, so we leave it out.
6 Vertical asymptote
The bottom: x − 3 = 0 when x = 3.
We can never divide by 0.
The top is not 0 there, so near it the graph shoots up or down: a vertical asymptote.
So the asymptotes are y = x + 1 and x = 3.
7 Check
Let's check with a picture.
The vertical asymptote x = 3, dashed.
The slant asymptote y = x + 1, dashed.
The graph of f: on each side it gets close to both dashed lines.
And with numbers: at x = 100, f is about 101.01, and the line gives 101.
Very close.
It works!
Let's look back.
First: compare the powers.
Then: set it up.
Then: first part of the answer.
Then: second part of the answer.
Then: read the slant asymptote.
Then: vertical asymptote.
Last: check.