Limit as x -> 2 of (x^3 - 8)/(x - 2) using L'Hôpital's rule Step by Step

Step by step

The question

Question: find the limit as x → 2 of (x³ − 8)/(x − 2).

lim x → 2 x³ − 8x − 2 =

1 Put the number in

Put 2 in place of x.

The top is 0, and the bottom is 0.

top: 2³ − 8 = 0

bottom: 2 − 2 = 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 3x².

The derivative of the bottom is 1.

The new limit is the same.

lim x → 2 x³ − 8x − 2 = lim x → 2 3x²1

3 Put the number in again

Put 2 in again.

Now the bottom is not 0.

So the limit is 12.

= 3(2)²1 = 121 = 12

4 Check

Let's check with a calculator, with x very close to 2.

x = 1.99 gives about 11.9401.

x = 1.99: x³ − 8x − 2 ≈ 11.9401

And x = 2.01 gives about 12.0601.

Both are close to 12.

Right!

x = 2.01: x³ − 8x − 2 ≈ 12.0601

Let's look back.

First: put the number in.

Then: use L'Hôpital's rule.

Then: put the number in again.

Last: check.

Answer