Unit 4 · Topic 4.7

4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

L'Hospital's Rule handles limits of quotients that come out as 0/0 or ∞/∞. If the limit of f(x)/g(x) has one of those forms, it equals the limit of f′(x)/g′(x), provided that new limit exists. You must show the indeterminate form before you use it.

Key terms

L'Hospital's Rule
indeterminate form
0/0
∞/∞

The rule and its conditions

Suppose f and g are differentiable near c (except possibly at c), and g′(x) ≠ 0 near c. If lim (x→c) f(x) = 0 and lim (x→c) g(x) = 0, or if both limits are ±∞, then lim (x→c) f(x)/g(x) = lim (x→c) f′(x)/g′(x), as long as the limit on the right exists (or is ±∞).

It also works for one-sided limits and for limits as x→∞ or x→−∞.

Why it makes sense: if f(c) = g(c) = 0, then near c local linearity (4.6) gives f(x) ≈ f′(c)(x − c) and g(x) ≈ g′(c)(x − c). Divide, and the (x − c) factors cancel, leaving about f′(c)/g′(c).

How to use it

A safe routine:

Substitute first. Write down the form you get, like 0/0.
If and only if it's 0/0 or ∞/∞, differentiate the top and bottom separately.
Take the new limit. If it is still 0/0 or ∞/∞, you may apply the rule again.
If substitution gives a real number or (nonzero)/0, don't use the rule.

Not the quotient rule

L'Hospital's Rule differentiates the numerator and denominator separately: f′(x)/g′(x). It does not use the quotient rule, which would compute the derivative of the whole fraction. That's a different thing entirely.

Growth rates

L'Hospital's Rule makes the growth-rate facts from 1.15 provable. lim (x→∞) x²/eˣ is ∞/∞, then 2x/eˣ is ∞/∞, then 2/eˣ → 0. So eˣ eventually beats any power of x. Similarly, lim (x→∞) ln x / √x = 0, so ln x grows slower than any positive power.

Using it with tables

Sometimes f and g are given only by values. If f(3) = g(3) = 0, f and g are differentiable, and f′ and g′ are continuous at 3 with g′(3) ≠ 0, then lim (x→3) f(x)/g(x) = f′(3)/g′(3). Exam questions often ask this, sometimes needing the rule twice using f″ and g″.

What you won't be tested on

Other indeterminate forms, like ∞ − ∞, 0·∞, and exponent forms such as 1 raised to the ∞ power, are not assessed on the AP exam. Only 0/0 and ∞/∞ are tested, so don't spend study time on the others.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Using the rule twice
Find lim (x→0) (eˣ − 1 − x)/x².
Show the solutionHide the solution
1. Step 1: Substitute: (1 − 1 − 0)/0 = 0/0. Indeterminate, so the rule applies.
2. Step 2: Differentiate top and bottom: lim (x→0) (eˣ − 1)/(2x). Substitute: 0/0 again.
3. Step 3: Apply the rule again: lim (x→0) eˣ/2.
4. Step 4: Substitute: 1/2.
Answer: 1/2
Example 2
An ∞/∞ limit at infinity
Find lim (x→∞) ln x / √x.
Show the solutionHide the solution
1. Step 1: As x→∞, ln x → ∞ and √x → ∞. Form: ∞/∞.
2. Step 2: Differentiate: (1/x) / (1/(2√x)) = 2√x/x = 2/√x.
3. Step 3: As x→∞, 2/√x → 0.
Answer: 0
Example 3
Trap: using the rule on a limit that isn't indeterminate
Find lim (x→0) cos x / (x + 1). A student uses L'Hospital's Rule and gets lim (x→0) (−sin x)/1 = 0. What went wrong?
Show the solutionHide the solution
1. Step 1: Substitute first: cos 0/(0 + 1) = 1/1 = 1.
2. Step 2: That's a real number, not 0/0 or ∞/∞, so L'Hospital's Rule doesn't apply. Using it anyway gives a wrong answer.
3. Step 3: The limit is simply 1, by direct substitution.
Answer: 1. The student's answer of 0 is wrong because the conditions of L'Hospital's Rule weren't met.

Common mistakes

Using L'Hospital's Rule without first showing 0/0 or ∞/∞. On free response, you lose the point if the form isn't shown.
Applying the quotient rule instead of differentiating top and bottom separately.
Applying the rule to a limit that isn't indeterminate, which can give a wrong answer.

On the exam

In free response, write the limits of the numerator and denominator (for example, “lim f(x) = 0 and lim g(x) = 0”) before applying the rule. This often appears with functions given by tables or as a step in a bigger question.
In multiple choice, look for limits that are 0/0 forms involving eˣ, ln x or trig, where the rule is faster than algebra.

Connected topics

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Check yourself

4 questions on 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms. Pick an answer to see if you got it, and why.

Question 1 of 4

What is lim (x→0) (e³ˣ − 1)/sin(2x)?

Question 2 of 4

What is lim (x→0) (1 − cos(3x))/x²?

Question 3 of 4

L'Hospital's Rule can be applied directly to which of the following limits? I. lim (x→0) (sin x)/(x + 1) II. lim (x→∞) eˣ/x² III. lim (x→1) (ln x)/(x − 1)

Question 4 of 4

What is lim (x→0) (eˣ − 1 − x)/x² ?

0 of 4 answered