L'Hôpital's Rule

On this placeholder screen we summarize how to use L'Hôpital's Rule to compute certain limits in Calculus, and provide free practice problems, each with a complete solution one click away so you can learn how to solve these and be exam-ready.

MATHENO ESSENTIALS: L'Hôpital's Rule

L'Hôpital's Rule:

I f l i m 𝑥 \to 𝑎 𝑓 (𝑥) 𝑔 (𝑥) \to 00 o r \infty \infty,

t h e n : l i m 𝑥 \to 𝑎 𝑓 (𝑥) 𝑔 (𝑥) = l i m 𝑥 \to 𝑎 𝑑 𝑑 𝑥 [𝑓 (𝑥)] 𝑑 𝑑 𝑥 [𝑔 (𝑥)]

That is, if you take a limit and it's in the form $00$ or $\infty \infty$ , then you can take the derivative of the numerator and the derivative of the denominator and find that limit instead.

$Tip icon$

Warning: Before taking the derivatives, verify that the original limit is in the form $00$ or $\infty \infty$ . Otherwise you cannot use L'Hôpital's Rule. (This is a favorite way to "trick" students on exams. Be careful!)

Practice Problem 1: sin & cos oft-used limits

We've been using these limits since the start of the semester. We're now easily going to prove that they're true.

(a)

Use L'Hôpital's Rule to show that

l i m 𝑥 \to 0 s i n 𝑥 𝑥 = 1

(b)

Use L'Hôpital's Rule to show that

l i m 𝑥 \to 0 c o s 𝑥 - 1 𝑥 = 0

Practice Problem 2: Find the limits

Find the requested limits.

(a)

l i m 𝑥 \to \infty 3 𝑥 2 - 5 𝑥 + 2 4 𝑥 2 - 6

(b)

l i m 𝑥 \to \infty 𝑥 2 (l n 𝑥) 2

(c)

l i m 𝑥 \to 2 𝑥 2 - 4 𝑥 + 2

(d)

l i m 𝑥 \to 0 𝑥 t a n 𝑥

(a) $34$
(b) $\infty$
(c)0 (cannot use L'Hôpital's Rule)
(d)1

[close solution]

Practice Problem 3: More find the limits

Evaluate the following limits:

(a)

l i m 𝑥 \to 0 𝑥 2 𝑒 𝑥

(b)

l i m 𝑥 \to 1 l n 𝑥 𝑥 - 1

(c)

l i m 𝑥 \to 0 5 𝑥 - 2 𝑥 𝑥

[Hint: Recall that

𝑑 𝑑 𝑥 𝑎 𝑥 = l n 𝑎 \cdot 𝑎 𝑥

]

(d)

l i m 𝑥 \to 0 c o s (2 𝑥) - c o s (𝑥) 𝑥

Limits in the form

0 𝑐 𝑑 𝑜 𝑡 𝑖 𝑛 𝑓 𝑡 𝑦

Often when you're asked about a limit that's initially in the form $0 𝑐 𝑑 𝑜 𝑡 𝑖 𝑛 𝑓 𝑡 𝑦$ , you can cleverly rewrite it in order to be able to use L'Hôpital's Rule. These questions illustrate.

Evaluate the following limits:

(a)

l i m 𝑥 \to 0 + 𝑥 l n 𝑥

(b)

l i m 𝑥 \to \infty 𝑥 2 𝑒 - 𝑥

(a)0
(b)0

[close solution]

Practice Problem #5: Limits in the form '

𝑖 𝑛 𝑓 𝑡 𝑦 - 𝑖 𝑛 𝑓 𝑡 𝑦

Sometimes when you're asked about a limit that's initially in the form ' $𝑖 𝑛 𝑓 𝑡 𝑦 - 𝑖 𝑛 𝑓 𝑡 𝑦,$ ' you can rewrite it in order to be able to use L'Hôpital's Rule. These questions illustrate.

(a)

l i m 𝑥 \to 0 (1 𝑥 - 1 s i n 𝑥)

(b)

l i m 𝑥 \to 0 (c o t 𝑥 - c s c 𝑥)

(c)

l i m 𝑥 \to \infty (\sqrt 𝑥 2 + 𝑥 - 𝑥)

(a)0
(b)0
(c) $12$

[close solution]

Practice Problem 6: Limits of the form

00

Missing open brace for superscript

, or

𝑖 𝑛 𝑓 𝑡 𝑦 0

If you encounter a limit in the form $00$ , $Missing open brace for superscript$ , or $𝑖 𝑛 𝑓 𝑡 𝑦 0$ , then:

Set $𝑦 =$ [the function you're given];
Take the $𝑙 𝑛$ of both sides of that equation;
Find the limit of $𝑙 𝑛 𝑦$ using L'Hôpital's Rule as necessary;
Return to the limit of the original function by recalling that $𝑦 = 𝑒 𝑙 𝑛 𝑦$ .

The following problems illustrate.

Find the requested limits:

(a)

l i m 𝑥 \to \infty 𝑥 1 / 𝑥

(b)

l i m 𝑥 \to 0 + 𝑥 𝑥

(c)

l i m 𝑥 \to 0 + (s i n 𝑥) t a n 𝑥

(d)

l i m 𝑥 \to \infty (1 + 1 𝑥) 𝑥

[Note: This is a famous limit that can be used to define

𝑒

(a)1
(b)1
(c)1
(d) $𝑒$

[close solution]

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L'Hôpital's Rule

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