Step by step
The question
Question: find the limit as x → −5 of (x² + 3x − 10)/(x + 5), using a table.
1 Try x = −5
First, try x = −5 itself.
The top is 0, and the bottom is 0.
We cannot divide by 0.
So f(x) has no value at −5.
But a limit does not need the value at −5.
It asks: as x gets close to −5, 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 −5: −5.1, −5.01, −5.001.
They come closer and closer to −5.
Put −5.1 in: we get −7.1.
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 −5: −4.9, −4.99, −4.999.
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 = −5.0001 gives −7.0001.
x = −4.9999 gives −6.9999.
Still close to −7.
Right!
Let's look back.
First: try x = −5.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.