Limit as x -> 1 of (x^2 - 1)/(x^2 + 3x

Step by step

The question

Question: find the limit as x → 1 of (x² − 1)/(x² + 3x − 4).

lim x → 1 x² − 1x² + 3x − 4 =

1 Put the number in

Put 1 in place of x.

The top is 0, and the bottom is 0.

1² − 11² + 3(1) − 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 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 x² − 1x² + 3x − 4 = lim x → 1 ~~(x − 1)~~(x + 1)~~(x − 1)~~(x + 4)

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

So it is (x − 1)(x + 1).

a² − b² = (a − b)(a + b), b = 1

We need two numbers that multiply to −4 and add to 3: 4 and −1.

4 × (−1) = −4, 4 + (−1) = 3

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: (x + 1)/(x + 4).

= lim x → 1 (x + 1)(x + 4)

4 Put the number in again

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

Put 1 in again.

= (1 + 1)(1 + 4)

So the limit is 2/5.

= 25

5 Check

Let's check with numbers close to 1.

x = 0.99 gives about 0.3988.

x = 0.99: f(x) ≈ 0.3988

x = 1.01 gives about 0.4012.

Both are close to 2/5.

Right!

x = 1.01: f(x) ≈ 0.4012

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer