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.

(−1)² − 1(−1) + 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: −1.1, −1.01, −1.001.

They come closer and closer to −1.

Put −1.1 in: we get −2.1.

f(−1.1) = (−1.1)² − 1(−1.1) + 1 = 1.21 − 1−1.1 + 1 = 0.21−0.1 = −2.1

−2.1

The others the same way, with a calculator.

−2.01

−2.001

From the left, f(x) gets close to −2.

from the left: f(x) → −2

3 From the right

Now numbers just above −1: −0.9, −0.99, −0.999.

Work out f(x) for each one.

−1.9

−1.99

−1.999

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 = −1.0001 gives −2.0001.

x = −1.0001: f(x) = −2.0001

x = −0.9999 gives −1.9999.

Still close to −2.

Right!

x = −0.9999: f(x) = −1.9999

Let's look back.

First: try x = −1.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer