Limit as x -> 1 of (x^2 - 1)/ln(x) Step by Step

Step by step

The question

Question: find the limit as x → 1 of (x² − 1) over ln x.

lim x → 1 x² − 1ln x =

1 Put the number in

Put 1 in place of x.

The top is 0, and the bottom is 0.

top: 1² − 1 = 0

bottom: ln 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 2x.

The derivative of the bottom is 1 over x.

The new limit is the same.

lim x → 1 x² − 1ln x = lim x → 1 2x1x

3 Put the number in again

Put 1 in again.

Now the bottom is not 0.

So the limit is 2.

= 2(1)11 = 21 = 2

4 Check

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

x = 0.99 gives about 1.98.

x = 0.99: x² − 1ln x ≈ 1.98

And x = 1.01 gives about 2.02.

Both are close to 2.

Right!

x = 1.01: x² − 1ln x ≈ 2.02

Let's look back.

First: put the number in.

Then: use L'Hôpital's rule.

Then: put the number in again.

Last: check.

Answer