Step by step
The question
Question: find the limit as x → 2 of (x² + x − 6)/(x − 2).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Put the number in
Put 2 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 2.
We find it.
2 Factor
x − 2 is the piece that is 0 at x = 2.
The top and the bottom are both 0 there, so both have (x − 2) in them.
We factor them.
To factor x² + x − 6, the rule: we need two numbers that multiply to −6 and add to 1.
They are 3 and −2: they multiply to −6 and add to 1.
Now we write the top and the bottom in factors.
3 Cancel
x gets close to 2, but it is never 2.
So x − 2 is not 0.
A number over itself is 1.
So we cross out (x − 2) on the top and the bottom.
Write what is left: x + 3.
4 Put the number in again
Now x − 2 is gone, so the bottom is not 0.
Put 2 in again.
So the limit is 5.
5 Check
Let's check with numbers close to 2.
x = 1.99 gives about 4.99.
x = 2.01 gives about 5.01.
Both are close to 5.
Right!
6 Pick the choice
Our answer is limit = 5.
That is choice B.
7 Why not the others
A is 0.
Wrong: put in 2 at once: 0/0 means simplify first.
That is the Careful mistake from before.
C is 4.
Wrong: put 2 into x² only.
D is 3.
Wrong: factored to (x + 1): the factors are (x + 3)(x − 2).
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.