Step by step
The question
Question: find the limit as x → −3 of (x² + 4x + 3)/(x + 3), using a table.
1 Try x = −3
First, try x = −3 itself.
The top is 0, and the bottom is 0.
We cannot divide by 0.
So f(x) has no value at −3.
But a limit does not need the value at −3.
It asks: as x gets close to −3, 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 −3: −3.1, −3.01, −3.001.
They come closer and closer to −3.
Put −3.1 in: we get −2.1.
The others the same way, with a calculator.
From the left, f(x) gets close to −2.
3 From the right
Now numbers just above −3: −2.9, −2.99, −2.999.
Work out f(x) for each one.
From the right, f(x) gets close to −2 too.
4 Read the limit
Both sides get close to the same number.
So the limit is −2.
5 Check
Let's check with numbers even closer.
x = −3.0001 gives −2.0001.
x = −2.9999 gives −1.9999.
Still close to −2.
Right!
Let's look back.
First: try x = −3.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.