Limit as x -> 2 of (x^2 - 4)/(x^2 + x

Step by step

The question

Question: find the limit as x → 2 of (x² − 4)/(x² + x − 6).

lim x → 2 x² − 4x² + x − 6 =

1 Put the number in

Put 2 in place of x.

The top is 0, and the bottom is 0.

2² − 42² + 2 − 6 = 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 2.

We find it.

Careful: not 00 = 0 ✗

2 Factor

When 0/0 comes out, (x − 2) hides in the top and the bottom.

Factor them.

lim x → 2 x² − 4x² + x − 6 = lim x → 2 ~~(x − 2)~~(x + 2)~~(x − 2)~~(x + 3)

x² − 4 is a difference of two squares: x² minus 2 squared.

So it is (x − 2)(x + 2).

a² − b² = (a − b)(a + b), b = 2

We need two numbers that multiply to −6 and add to 1: 3 and −2.

3 × (−2) = −6, 3 + (−2) = 1

3 Cancel

x gets close to 2, but it is never 2.

So x − 2 is not 0.

x ≠ 2, so x − 2 ≠ 0

A number over itself is 1.

So we cross out (x − 2) on the top and the bottom.

Write what is left: (x + 2)/(x + 3).

= lim x → 2 (x + 2)(x + 3)

4 Put the number in again

Now x − 2 is gone, so the bottom is not 0.

Put 2 in again.

= (2 + 2)(2 + 3)

So the limit is 4/5.

= 45

5 Check

Let's check with numbers close to 2.

x = 1.99 gives about 0.7996.

x = 1.99: f(x) ≈ 0.7996

x = 2.01 gives about 0.8004.

Both are close to 4/5.

Right!

x = 2.01: f(x) ≈ 0.8004

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer