Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
Check an indeterminate quotient and theorem hypotheses before comparing numerator and denominator derivatives.
College Board Topic 4.7 · CED p. 88 (PDF p. 93) · CED effective Fall 2020; project course outline 2026–27.
A fraction whose numerator and denominator both approach zero has form 0/0. A fraction whose numerator and denominator both grow without bound in magnitude has the corresponding infinite-over-infinite form. These forms do not name the answer. They say that the two competing behaviors still need investigation. The AP-required scope here is these quotient forms; other indeterminate forms are optional extensions.
L’Hospital’s Rule compares the rates at which the numerator and denominator change. For a quotient f(x)/g(x), first verify a required indeterminate form. On an appropriate punctured one-sided neighborhood, the numerator and denominator must be differentiable and g′ must be nonzero; if the limit of f′/g′ exists (including an infinite limit), the original quotient has that limit. One-sided limits and limits at infinity can also qualify. You differentiate numerator and denominator separately; you are not differentiating the quotient itself.
For lim as x → 0 of (eˣ − 1)/x, substitution gives 0/0 and the derivative quotient eˣ/1 tends to 1. For lim as x → ∞ of ln x/x, the infinite-over-infinite form gives (1/x)/1 → 0. Algebra or familiar limits may be quicker than this rule. If a first application leaves an indeterminate form, recheck conditions before applying it again.
Never apply the rule to 1/0 merely because the denominator vanishes. For (x + 1)/x as x → 0⁺, the original quotient tends to +∞, whereas its derivative quotient equals 1. Also, a derivative quotient with no limit makes the rule inconclusive; it does not by itself prove that the original quotient has no limit. A graph or calculator estimate can support a conjecture but cannot replace the form and condition checks.
Let's work through it
Find lim as x → 0 of (1 − cos x)/x². Direct substitution gives 0/0.
Why: L’Hospital’s Rule requires a qualifying indeterminate quotient.
Differentiate separately to obtain sin x/(2x), still 0/0.
Why: The rule preserves the limit here because the functions are differentiable and the denominator derivative is nonzero for x ≠ 0 near zero.
Recheck and differentiate separately again: cos x/2 → 1/2.
Why: The second quotient again meets the rule’s conditions and now has a directly evaluable limit.
The original limit is 1/2.
Why: Two justified applications connect the original quotient to the final limit.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Is 7/0 an indeterminate form covered by this rule?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Is 7/0 an indeterminate form covered by this rule?
Worked answer and reason
No.
A vanishing denominator alone is insufficient.
Retrieval 2
Does L’Hospital’s Rule use the quotient rule?
Worked answer and reason
No: use the limit of f′/g′ after checking the hypotheses.
The theorem compares two derivative functions, not the derivative of the quotient.
Retrieval 3
If f′/g′ has no limit, must f/g have no limit?
Worked answer and reason
No; this version of the rule is inconclusive.
Failure of a theorem’s sufficient condition is not the opposite conclusion.
Catch the tempting mistake
- 0/0 means the limit is zero.
- It is an indeterminate form: different quotients of that form have different limits.
- Differentiate numerator and denominator whenever substitution fails.
- Identify the form and verify the theorem’s conditions first.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 4 problem 26 · No calculator · analytic
- Unit 4 problem 27 · No calculator · analytic
- Unit 4 problem 28 · No calculator · analytic
- Unit 4 problem 29 · No calculator · verbal
- Unit 4 problem 30 · No calculator · analytic
For the teacher
- Use a mixed set of direct substitution, factoring, familiar limits, and L’Hospital cases; require learners to choose a method.
- A calculator zoom can miss oscillation or unstable subtraction. Use it for checking, with analytic justification as the final authority.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Indeterminate form
An indeterminate form is a pattern in a proposed limit that does not by itself determine the limit’s value. The forms 0/0 and infinity divided by infinity can lead to several different outcomes. For example, x/x approaches 1 as x approaches 0, while x²/x approaches 0 despite the same 0/0 substitution pattern. The form signals that a justified further analysis is needed.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.