Limit as x -> -1 of (2x^2 - 2)/(x + 1) Step by Step

Step by step

The question

Question: find the limit as x → −1 of (2x² − 2)/(x + 1).

lim x → −1 2x² − 2x + 1 =

1 Put the number in

Put −1 in place of x.

The top is 0, and the bottom is 0.

2(−1)² − 2(−1) + 1 = 00

0/0 has no value.

It does not tell us the limit.

It tells us to work more.

00: no value yet. Work more!

Careful!

0/0 is not 0, and it is not 1.

Some piece of the top and the bottom is 0 at −1.

We find it.

Careful: not 00 = 0 ✗

2 Factor

When 0/0 comes out, (x + 1) hides in the top and the bottom.

Factor them.

lim x → −1 2x² − 2x + 1 = lim x → −1 2~~(x + 1)~~(x − 1)~~(x + 1)~~

First take out the 2 that is in every term.

x² − 1 is a difference of two squares: x² minus 1 squared.

So it is (x − 1)(x + 1).

2x² − 2 = 2(x² − 1)

a² − b² = (a − b)(a + b), b = 1

3 Cancel

x gets close to −1, but it is never −1.

So x + 1 is not 0.

x ≠ −1, so x + 1 ≠ 0

A number over itself is 1.

So we cross out (x + 1) on the top and the bottom.

Write what is left: 2(x − 1).

= lim x → −1 2(x − 1)

4 Put the number in again

Now x + 1 is gone, so the bottom is not 0.

Put −1 in again.

= 2(−1 − 1)

= 2 × (−2)

So the limit is −4.

= −4

5 Check

Let's check with numbers close to −1.

x = −1.01 gives about −4.02.

x = −1.01: f(x) ≈ −4.02

x = −0.99 gives about −3.98.

Both are close to −4.

Right!

x = −0.99: f(x) ≈ −3.98

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer