Step by step
The question
Question: find the limit as x → 3 of (x² + x − 12)/(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: 2.9, 2.99, 2.999.
They come closer and closer to 3.
Put 2.9 in: we get 6.9.
The others the same way, with a calculator.
From the left, f(x) gets close to 7.
3 From the right
Now numbers just above 3: 3.1, 3.01, 3.001.
Work out f(x) for each one.
From the right, f(x) gets close to 7 too.
4 Read the limit
Both sides get close to the same number.
So the limit is 7.
5 Check
Let's check with numbers even closer.
x = 2.9999 gives 6.9999.
x = 3.0001 gives 7.0001.
Still close to 7.
Right!
Let's look back.
First: try x = 3.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.