Step by step
The question
Question: find the slant asymptote of f(x) = (x² − 3x + 1)/(x − 1).
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 − 1 goes outside, x² − 3x + 1 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 − 1) is x² − x.
We write it under, with a minus: we take it away.
x² take away x² is 0.
That is why we chose x.
−3x take away −x is −2x.
Bring down the next part: 1.
4 Second part of the answer
Divide the first parts: −2x divided by x is −2.
We write it on top.
Multiply: −2 times (x − 1) is −2x + 2.
We write it under, with a minus: we take it away.
−2x take away −2x is 0.
That is why we chose −2.
1 take away 2 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 − 2, minus 1/(x − 1).
5 Read the slant asymptote
When x is big, 1/(x − 1) is tiny.
At x = 100 it is 1/99: about 0.01.
Almost 0.
So for a big x, f(x) is close to x − 2.
The slant asymptote is y = x − 2.
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 − 1 = 0 when x = 1.
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 − 2 and x = 1.
7 Check
Let's check with a picture.
The vertical asymptote x = 1, dashed.
The slant asymptote y = x − 2, dashed.
The graph of f: on each side it gets close to both dashed lines.
And with numbers: at x = 100, f is about 97.99, and the line gives 98.
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.