Limit as x -> 3 of (x^2 - 9)/(x^2 - x

Step by step

The question

Question: find the limit as x → 3 of (x² − 9)/(x² − x − 6).

lim x → 3 x² − 9x² − x − 6 =

1 Put the number in

Put 3 in place of x.

The top is 0, and the bottom is 0.

3² − 93² − 3 − 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 3.

We find it.

Careful: not 00 = 0 ✗

2 Factor

When 0/0 comes out, (x − 3) hides in the top and the bottom.

Factor them.

lim x → 3 x² − 9x² − x − 6 = lim x → 3 ~~(x − 3)~~(x + 3)~~(x − 3)~~(x + 2)

x² − 9 is a difference of two squares: x² minus 3 squared.

So it is (x − 3)(x + 3).

a² − b² = (a − b)(a + b), b = 3

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

2 × (−3) = −6, 2 + (−3) = −1

3 Cancel

x gets close to 3, but it is never 3.

So x − 3 is not 0.

x ≠ 3, so x − 3 ≠ 0

A number over itself is 1.

So we cross out (x − 3) on the top and the bottom.

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

= lim x → 3 (x + 3)(x + 2)

4 Put the number in again

Now x − 3 is gone, so the bottom is not 0.

Put 3 in again.

= (3 + 3)(3 + 2)

So the limit is 6/5.

= 65

5 Check

Let's check with numbers close to 3.

x = 2.99 gives about 1.2004.

x = 2.99: f(x) ≈ 1.2004

x = 3.01 gives about 1.1996.

Both are close to 6/5.

Right!

x = 3.01: f(x) ≈ 1.1996

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer