Step by step
The question
Question: find the limit as x → −3 of (x² + x − 6)/(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.
We need two numbers that multiply to −6 and add to 1: 3 and −2.
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 − 2.
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 −5.
5 Check
Let's check with numbers close to −3.
x = −3.01 gives about −5.01.
x = −2.99 gives about −4.99.
Both are close to −5.
Right!
Let's look back.
First: put the number in.
Then: factor.
Then: cancel.
Then: put the number in again.
Last: check.