Limit as x -> infinity of ln(x)/x

Step by step

The question

Question: find the limit as x → ∞ of ln x over x.

1 Let x get huge

Let x get huge.

ln x gets huge, and so does x.

Infinity over infinity tells us nothing yet.

Which one wins?

We cannot see it yet.

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.

Careful!

This is not the quotient rule.

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

The derivative of the top is 1 over x.

The derivative of the bottom is 1.

The new limit is the same.

3 Let x get huge

Let x get huge now.

The top goes to 0, and the bottom goes to 1.

So the limit is 0.

4 Check

Let's check with a calculator: x = 10 gives about 0.2303.

And x = 20 gives about 0.1498.

Both are close to 0.

Right!

Let's look back.

First: let x get huge.

Then: use L'Hôpital's rule.

Then: let x get huge.

Last: check.