Step by step
The question
Question: find the limit as x → 1 of (x² + 2x − 3)/(x − 1).
1 Put the number in
Put 1 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 1.
We find it.
2 Factor
When 0/0 comes out, (x − 1) hides in the top and the bottom.
Factor them.
We need two numbers that multiply to −3 and add to 2: 3 and −1.
3 Cancel
x gets close to 1, but it is never 1.
So x − 1 is not 0.
A number over itself is 1.
So we cross out (x − 1) on the top and the bottom.
Write what is left: x + 3.
4 Put the number in again
Now x − 1 is gone, so the bottom is not 0.
Put 1 in again.
So the limit is 4.
5 Check
Let's check with numbers close to 1.
x = 0.99 gives about 3.99.
x = 1.01 gives about 4.01.
Both are close to 4.
Right!
Let's look back.
First: put the number in.
Then: factor.
Then: cancel.
Then: put the number in again.
Last: check.