L'Hôpital's Rule

Some limits resist every technique in this chapter. doesn’t factor, has no conjugate and isn’t a squeeze. L’Hôpital’s rule handles it, and most other and limits, by replacing the functions with their derivatives. It uses derivatives, which the next chapter develops fully, so a short table of the ones you need is included below.

What L’Hôpital’s Rule Says

Suppose gives or when you substitute. If and are differentiable near , and there (except possibly at ), then

as long as the right-hand limit exists (or is ). The same holds for one-sided limits and for . The key word is separately: differentiate the top on its own and the bottom on its own. This is not the quotient rule.

The Derivatives You’ll Need

Derivatives of sums are sums of derivatives, and constants multiply through. That’s enough for every example on this page.

Why It Works

If , then close to each function is almost its tangent line: and . The factors cancel in the ratio, leaving . In the picture, starts out like and like , so their ratio heads for 2.

When to Use L’Hôpital’s Rule

Always substitute first. The rule applies only to and . Using it anywhere else gives wrong answers: by substitution, but “differentiating” would give .

How to Use L’Hôpital’s Rule, Step by Step

1. Substitute the value approaches. If you get a number, stop: that’s the limit.
2. Identify the form. If it’s or , continue. If it’s , or an exponent form, rewrite it as a fraction first (see below).
3. Differentiate the numerator and the denominator separately.
4. Take the limit of the new fraction. If it’s still indeterminate, go back to step 3.
5. Sanity-check the answer with a value close to , as you would for any limit.

Example 1: A 0/0 Limit

Find .

Substitution gives . Differentiate top and bottom: . (The squeeze theorem proved this limit without derivatives; that proof is needed to find the derivative of in the first place, so the two results don’t go in a circle.)

Example 2: Checking Against Factoring

Find .

, so . Evaluating limits gets the same answer by factoring. For simple algebra, factoring is often just as quick; the rule shines when algebra can’t help.

Example 3: An ∞/∞ Limit

Find .

. One application gives , still . A second gives , matching the degree rule from limits at infinity.

Example 4: Applying the Rule Twice

Find .

, so take derivatives: . Still , so again: . Re-check the form before each step; stop as soon as substitution gives a number.

Example 5: Comparing Growth Rates

: , then , then . Similarly . These prove the growth ranking from the previous lessons: logarithms grow slower than any power of , and powers grow slower than .

Other Indeterminate Forms

. : rewrite as , which is . The rule gives .

. : combine into , which is . Two applications of the rule give 0.

. : let be the expression and take logs, , a form whose limit is 1. So . This is the limit that defines , and it’s the same idea behind continuous compound interest.

When the Rule Doesn’t Help

Occasionally the rule goes in circles. For , each application swaps the fraction for its reciprocal and never ends. Dividing top and bottom by instead gives the answer, 1, straight away. And if doesn’t exist, the rule says nothing about the original limit; try another method.

A Short History

The rule carries the name of Guillaume de l’Hôpital, a French nobleman who published it in 1696 in Analyse des infiniment petits, the first textbook on differential calculus. The mathematics, though, came from his tutor Johann Bernoulli, whom l’Hôpital paid for his discoveries. You’ll see the name spelled “L’Hôpital”, “L’Hospital” and “Lhopital”; they’re all the same rule.

Common Mistakes to Avoid

• Using the quotient rule instead of differentiating top and bottom separately.
• Applying the rule without checking the form. It’s only for and .
• Forgetting to re-check after each application; you may already have a number.
• Skipping the rewrite for , and exponent forms.

Practice Problems

Work each one before opening the answer.

Problem 1. Find .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. Can L’Hôpital’s rule be used for ?

That completes the limits chapter. With limits in place, Calculus 1 moves on to derivatives: the limit of the average rate of change, and the rules that make finding it quick.