Step by step
The question
Question: find the asymptotes and holes of f(x) = 5/(x − 3).
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 − 3 = 0 when x = 3.
Look: at x = 3.1, the bottom is tiny.
So f(x) is about 50: 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 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 0.05.
Close to 0.
So the horizontal asymptote is y = 0.
So: x = 3 and y = 0.
3 Check
Let's check: put x = 3 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: vertical asymptotes.
Then: horizontal asymptote.
Last: check.