Limit as x -> 1 of (x^4 - 1)/(x^2 - 1) using L'Hôpital's rule Step by Step

Step by step

The question

Question: find the limit as x → 1 of (x⁴ − 1)/(x² − 1).

lim x → 1 x⁴ − 1x² − 1 =

1 Put the number in

Put 1 in place of x.

The top is 0, and the bottom is 0.

top: 1⁴ − 1 = 0

bottom: 1² − 1 = 0

0/0 has no value.

It tells us nothing yet.

This is where L'Hôpital's rule helps.

form 00

2 Use L'Hôpital's rule

The rule: when we get 0/0, or infinity over infinity, we may take the derivative of the top, and the derivative of the bottom.

The new limit is the same.

lim x → a f(x)g(x) = lim x → a f′(x)g′(x)

Careful!

This is not the quotient rule.

The top gets its own derivative, and the bottom gets its own derivative.

Careful: not the quotient rule: top and bottom each alone

The derivative of the top is 4x³.

The derivative of the bottom is 2x.

The new limit is the same.

lim x → 1 x⁴ − 1x² − 1 = lim x → 1 4x³2x

3 Put the number in again

Put 1 in again.

Now the bottom is not 0.

So the limit is 2.

= 4(1)³2(1) = 42 = 2

4 Check

Let's check with a calculator, with x very close to 1.

x = 0.99 gives about 1.9801.

x = 0.99: x⁴ − 1x² − 1 ≈ 1.9801

And x = 1.01 gives about 2.0201.

Both are close to 2.

Right!

x = 1.01: x⁴ − 1x² − 1 ≈ 2.0201

Let's look back.

First: put the number in.

Then: use L'Hôpital's rule.

Then: put the number in again.

Last: check.

Answer