Step by step
The question
Question: find the limit as x → −1 of (x² − x − 2)/(x + 1), using a table.
1 Try x = −1
First, try x = −1 itself.
The top is 0, and the bottom is 0.
We cannot divide by 0.
So f(x) has no value at −1.
But a limit does not need the value at −1.
It asks: as x gets close to −1, what number does f(x) get close to?
Careful!
0/0 does not mean there is no limit.
It only means we must look closer.
2 From the left
Take numbers just below −1: −1.1, −1.01, −1.001.
They come closer and closer to −1.
Put −1.1 in: we get −3.1.
The others the same way, with a calculator.
From the left, f(x) gets close to −3.
3 From the right
Now numbers just above −1: −0.9, −0.99, −0.999.
Work out f(x) for each one.
From the right, f(x) gets close to −3 too.
4 Read the limit
Both sides get close to the same number.
So the limit is −3.
5 Check
Let's check with numbers even closer.
x = −1.0001 gives −3.0001.
x = −0.9999 gives −2.9999.
Still close to −3.
Right!
Let's look back.
First: try x = −1.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.