Limit as x -> -2 of (x^2 + 5x + 6)/(x^2

Step by step

The question

Question: find the limit as x → −2 of (x² + 5x + 6)/(x² − 4).

lim x → −2 x² + 5x + 6x² − 4 =

1 Put the number in

Put −2 in place of x.

The top is 0, and the bottom is 0.

(−2)² + 5(−2) + 6(−2)² − 4 = 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 + 5x + 6x² − 4 = lim x → −2 ~~(x + 2)~~(x + 3)~~(x + 2)~~(x − 2)

We need two numbers that multiply to 6 and add to 5: 3 and 2.

3 × 2 = 6, 3 + 2 = 5

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

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 + 3)/(x − 2).

= lim x → −2 (x + 3)(x − 2)

4 Put the number in again

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

Put −2 in again.

= (−2 + 3)(−2 − 2)

= 1−4

So the limit is −1/4.

= −14

5 Check

Let's check with numbers close to −2.

x = −2.01 gives about −0.2469.

x = −2.01: f(x) ≈ −0.2469

x = −1.99 gives about −0.2531.

Both are close to −1/4.

Right!

x = −1.99: f(x) ≈ −0.2531

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer