Step by step
The question
Question: find the limit as x → 1 of (x² + 2x − 3)/(x² − 1).
There are four choices.
First we solve it, the normal way.
Then we pick.
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
x − 1 is the piece that is 0 at x = 1.
The top and the bottom are both 0 there, so both have (x − 1) in them.
We factor them.
To factor x² + 2x − 3, the rule: we need two numbers that multiply to −3 and add to 2.
They are 3 and −1: they multiply to −3 and add to 2.
1 is 1 times 1.
So x² − 1 is x² minus 1²: one square take away another square.
The rule: a² − b² = (a − b)(a + b).
So it is (x − 1)(x + 1).
Now we write the top and the bottom in factors.
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)/(x + 1).
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 2.
5 Check
Let's check with numbers close to 1.
x = 0.99 gives about 2.005.
x = 1.01 gives about 1.995.
Both are close to 2.
Right!
6 Pick the choice
Our answer is limit = 2.
That is choice B.
7 Why not the others
A is 0.
Wrong: put in 1 at once: 0/0 means simplify first.
That is the Careful mistake from before.
C is 1.
Wrong: cancelled the x²: the factors are (x + 3)(x − 1) and (x + 1)(x − 1).
D is 4.
Wrong: lost the + 1 of the bottom.
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.