Step by step
The question
Question: find the limit as x → 3 of (x² − 6x + 9)/(x − 3).
1 Put the number in
Put 3 in place of x.
The top is 0, and the bottom is 0.
0/0 has no value.
It does not tell us the limit.
It tells us to work more.
Careful!
0/0 is not 0, and it is not 1.
Some piece of the top and the bottom is 0 at 3.
We find it.
2 Factor
When 0/0 comes out, (x − 3) hides in the top and the bottom.
Factor them.
Two numbers that multiply to 9 and add to −6: −3 and −3, the same number twice.
3 Cancel
x gets close to 3, but it is never 3.
So x − 3 is not 0.
A number over itself is 1.
So we cross out (x − 3) on the top and the bottom.
Write what is left: x − 3.
4 Put the number in again
Now x − 3 is gone, so the bottom is not 0.
Put 3 in again.
So the limit is 0.
5 Check
Let's check with numbers close to 3.
x = 2.99 gives about −0.01.
x = 3.01 gives about 0.01.
Both are close to 0.
Right!
Let's look back.
First: put the number in.
Then: factor.
Then: cancel.
Then: put the number in again.
Last: check.