Step by step
The question
Question: find the limit as x → 5 of (x² − 25)/(x − 5).
1 Put the number in
Put 5 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 5.
We find it.
2 Factor
When 0/0 comes out, (x − 5) hides in the top and the bottom.
Factor them.
x² − 25 is a difference of two squares: x² minus 5 squared.
So it is (x − 5)(x + 5).
3 Cancel
x gets close to 5, but it is never 5.
So x − 5 is not 0.
A number over itself is 1.
So we cross out (x − 5) on the top and the bottom.
Write what is left: x + 5.
4 Put the number in again
Now x − 5 is gone, so the bottom is not 0.
Put 5 in again.
So the limit is 10.
5 Check
Let's check with numbers close to 5.
x = 4.99 gives about 9.99.
x = 5.01 gives about 10.01.
Both are close to 10.
Right!
Let's look back.
First: put the number in.
Then: factor.
Then: cancel.
Then: put the number in again.
Last: check.