Step by step
The question
Question: find the limit as x → ∞ of x² over eˣ.
1 Let x get huge
Let x get huge.
x² gets huge, and so does eˣ.
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 2x.
The derivative of the bottom is eˣ.
The new limit is the same.
3 Use it again
Again, let x get huge.
2x gets huge, and so does eˣ.
Infinity over infinity tells us nothing yet.
Use the rule again.
The derivative of the top is 2.
The derivative of the bottom is eˣ.
The new limit is the same.
4 Let x get huge
Let x get huge now.
The top goes to 2, and the bottom goes to infinity.
So the limit is 0.
A number over something huge is almost 0.
5 Check
Let's check with a calculator: x = 10 gives about 0.0045.
And x = 20 gives about 8.2 × 10⁻⁷.
Both are close to 0.
Right!
Let's look back.
First: let x get huge.
Then: use L'Hôpital's rule.
Then: use it again.
Then: let x get huge.
Last: check.