Limit of (x^2 - 1)/(x - 1) as x -> 1 using a table Step by Step

Step by step

The question

Question: find the limit as x → 1 of (x² − 1)/(x − 1), using a table.

lim x → 1 x² − 1x − 1 =

1 Try x = 1

First, try x = 1 itself.

The top is 0, and the bottom is 0.

1² − 11 − 1 = 00

We cannot divide by 0.

So f(x) has no value at 1.

00: no value

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?

limit: the number f(x) gets close to

Careful!

0/0 does not mean there is no limit.

It only means we must look closer.

Careful: not 00 → no limit ✗

2 From the left

Take numbers just below 1: 0.9, 0.99, 0.999.

They come closer and closer to 1.

Put 0.9 in: we get 1.9.

f(0.9) = (0.9)² − 1(0.9) − 1 = 0.81 − 10.9 − 1 = −0.19−0.1 = 1.9

1.9

The others the same way, with a calculator.

1.99

1.999

From the left, f(x) gets close to 2.

from the left: f(x) → 2

3 From the right

Now numbers just above 1: 1.1, 1.01, 1.001.

Work out f(x) for each one.

2.1

2.01

2.001

From the right, f(x) gets close to 2 too.

from the right: f(x) → 2

4 Read the limit

Both sides get close to the same number.

So the limit is 2.

lim x → 1 x² − 1x − 1 = 2

5 Check

Let's check with numbers even closer.

x = 0.9999 gives 1.9999.

x = 0.9999: f(x) = 1.9999

x = 1.0001 gives 2.0001.

Still close to 2.

Right!

x = 1.0001: f(x) = 2.0001

Let's look back.

First: try x = 1.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer