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C16 · Unit C4 Contextual derivatives

L’Hospital’s rule and eligible forms

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 4.7
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

L’Hospital’s rule can evaluate 0 over 0 or ∞/∞ limits under its differentiability and derivative-ratio hypotheses. First identify the form; then differentiate numerator and denominator separately and reevaluate. This is a theorem about limits, not a formula for differentiating a quotient. Products and differences may require algebraic rewriting before the rule applies.

Worked example

lim as x tends to 0 of (eˣ minus 1)/x has form 0 over 0. The derivative ratio eˣ/1 tends to 1, so the original limit is 1. In contrast, lim (x+1)/(x+2) at zero is simply 1 over 2; applying L’Hospital there would incorrectly give 1.

Explore the mathematics

L’Hospital’s rule and eligible forms

Check an eligible indeterminate form before taking a derivative ratio.

Setting 3 of 5

Scroll sideways to see the full diagram.

Direct substitution gives 0/0. Does that determine the limit?xy123limₓ→₀ sin(3x)/x, angles in radians

Predict before continuing

Direct substitution gives 0/0. Does that determine the limit?

Read the complete explanation
  1. Direct substitution gives 0/0. Does that determine the limit?
  2. Check the hypotheses for L’Hospital: a suitable punctured interval, differentiable numerator and denominator, and an eligible indeterminate form.
  3. The derivative ratio is 3cos(3x)/1; its limit is 3.
  4. Hence this limit is 3. L’Hospital differentiates numerator and denominator separately; it does not apply the quotient rule to the ratio.
See the idea: L’Hospital’s rule and eligible forms
See the idea: L’Hospital’s rule and eligible formsThe quotient (eˣ−1)/x is undefined at zero, shown by the open point, but approaches 1. Its 0/0 form permits L’Hospital’s rule under the usual differentiability and derivative-ratio conditions.xy-10-0.50.5010.51.512limit 1

The quotient (eˣ−1)/x is undefined at zero, shown by the open point, but approaches 1. Its 0/0 form permits L’Hospital’s rule under the usual differentiability and derivative-ratio conditions.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not use the rule on a determinate form. When repeated use is needed, recheck eligibility at each step.

Investigate and explain

Sort a set of limits into substitution, algebra, squeeze and L’Hospital candidates. Explain the eligibility decision before calculating.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.