Step by step
The question
Question: find the limit as x → 3 of (x² − 9)/(x − 3).
There are four choices.
First we solve it, the normal way.
Then we pick.
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
x − 3 is the piece that is 0 at x = 3.
The top and the bottom are both 0 there, so both have (x − 3) in them.
We factor them.
9 is 3 times 3.
So x² − 9 is x² minus 3²: one square take away another square.
The rule: a² − b² = (a − b)(a + b).
So it is (x − 3)(x + 3).
Now we write the top and the bottom in factors.
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 6.
5 Check
Let's check with numbers close to 3.
x = 2.99 gives about 5.99.
x = 3.01 gives about 6.01.
Both are close to 6.
Right!
6 Pick the choice
Our answer is limit = 6.
That is choice A.
7 Why not the others
B is 3.
Wrong: cancelled to x and lost the + 3.
C is 9.
Wrong: put 3 into x² only.
D is 0.
Wrong: put in 3 at once: 0/0 means simplify first.
That is the Careful mistake from before.
Let's look back.
First: put the number in.
Then: factor.
Then: cancel.
Then: put the number in again.
Then: check.
Then: pick the choice.
Last: why not the others.