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:

If lim𝑥→𝑎𝑓(𝑥)𝑔(𝑥)→00 or ∞∞, then: lim𝑥→𝑎𝑓(𝑥)𝑔(𝑥)=lim𝑥→𝑎𝑑𝑑𝑥[𝑓(𝑥)]𝑑𝑑𝑥[𝑔(𝑥)]

That is, if you take a limit and it's in the form  00  or  ∞∞,  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  ∞∞. 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 lim𝑥→0sin⁡𝑥𝑥=1.
(b)
Use L'Hôpital's Rule to show that lim𝑥→0cos⁡𝑥−1𝑥=0.
Practice Problem 2: Find the limits

Find the requested limits.

(a)
lim𝑥→∞3𝑥2−5𝑥+24𝑥2−6
(b)
lim𝑥→∞𝑥2(ln⁡𝑥)2
(c)
lim𝑥→2𝑥2−4𝑥+2
(d)
lim𝑥→0𝑥tan⁡𝑥
Practice Problem 3: More find the limits

Evaluate the following limits:

(a)
lim𝑥→0𝑥2𝑒𝑥
(b)
lim𝑥→1ln⁡𝑥𝑥−1
(c)
lim𝑥→05𝑥−2𝑥𝑥 [Hint: Recall that 𝑑𝑑𝑥𝑎𝑥 =ln⁡𝑎 ⋅𝑎𝑥]
(d)
lim𝑥→0cos⁡(2𝑥)−cos⁡(𝑥)𝑥
Limits in the form 0 ⋅∞

Often when you're asked about a limit that's initially in the form 0 ⋅∞, you can cleverly rewrite it in order to be able to use L'Hôpital's Rule. These questions illustrate.

Evaluate the following limits:

(a)
lim𝑥→0+𝑥ln⁡𝑥
(b)
lim𝑥→∞𝑥2𝑒−𝑥
Practice Problem #5: Limits in the form '∞ −∞'

Sometimes when you're asked about a limit that's initially in the form '∞ −∞,' you can rewrite it in order to be able to use L'Hôpital's Rule. These questions illustrate.

(a)
lim𝑥→0(1𝑥−1sin⁡𝑥)
(b)
lim𝑥→0(cot⁡𝑥−csc⁡𝑥)
(c)
lim𝑥→∞(√𝑥2+𝑥−𝑥)
Practice Problem 6: Limits of the form 00, 1∞, or ∞0

If you encounter a limit in the form 00, 1∞, or ∞0, then:

  1. Set 𝑦 =[the function you're given];
  2. Take the ln of both sides of that equation;
  3. Find the limit of ln⁡𝑦 using L'Hôpital's Rule as necessary;
  4. Return to the limit of the original function by recalling that 𝑦 =𝑒ln⁡𝑦.

The following problems illustrate.

Find the requested limits:

(a)
lim𝑥→∞𝑥1/𝑥
(b)
lim𝑥→0+𝑥𝑥
(c)
lim𝑥→0+(sin⁡𝑥)tan⁡𝑥
(d)
lim𝑥→∞(1+1𝑥)𝑥 [Note: This is a famous limit that can be used to define 𝑒.]

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