Step by step
The question
Question: find the asymptotes and holes of f(x) = (x − 1)/(x − 1)(x + 2).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 Find the holes
Now look for the same bracket on the top and the bottom.
(x − 1) is on the top and the bottom.
(x − 1)/(x − 1) is 1.
Cross them out.
What is left: 1/(x + 2).
But the x that made the crossed bracket 0 is still not allowed in the question.
The graph just has one missing point there: a hole.
Careful!
A crossed-out bracket gives a hole, not an asymptote.
The graph only misses one point there.
Put x = 1 into what is left: y is 1/3.
So the hole is at (1, 1/3).
2 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
What is left at the bottom: x + 2 = 0 when x = −2.
Look: at x = −1.9, the bottom is tiny.
So f(x) is about 10: huge.
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 0.01.
Close to 0.
So the horizontal asymptote is y = 0.
So: hole (1, 1/3), 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: find the holes.
Then: vertical asymptotes.
Then: horizontal asymptote.
Last: check.