Step by step
The question
Question: find the asymptotes and holes of f(x) = (x − 4)/(x + 2).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 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 −59: huge.
2 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 1.
Look: at x = 100, f(x) is about 0.94.
Close to 1.
So the horizontal asymptote is y = 1.
So: x = −2 and y = 1.
3 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 1.
It works!
Let's look back.
First: vertical asymptotes.
Then: horizontal asymptote.
Last: check.