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

Syllabus
2020
Topic
4.7
Level
—

Learning objectives

—LIM-4.A—Determine limits of functions that result in indeterminate forms• LIM-4.A Determine limits of functions that result in indeterminate forms.• LIM-4.A.1 When the ratio of two functions tends to 0/0 or ∞/∞ in the limit, such forms are said to be indeterminate.- Exclusion statement: There are many other indeterminate forms, such as ∞ − ∞, for example, but these will not be assessed on either the AP Calculus AB or BC Exam. However, teachers may include these topics, if time permits.• LIM-4.A.2 Limits of the indeterminate forms 0/0 or ∞/∞ may be evaluated using L’Hospital’s Rule.• Enduring understanding LIM-4: L’Hospital’s Rule allows us to determine the limits of some indeterminate forms.

Check the Form Before Using L’Hospital’s Rule

A quotient approaching 0/00/0 or ∞/∞\infty/\infty is indeterminate: those symbols do not give the limit. L’Hospital’s Rule may replace the quotient with the quotient of its derivatives when the relevant derivatives exist and the new quotient has a limit.

\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}

  1. Substitute or analyze the numerator and denominator limits.\n2. Continue only if the quotient has form 0/00/0 or ∞/∞\infty/\infty.\n3. Differentiate the numerator and denominator separately.\n4. Evaluate the new quotient limit; if it is still one of the allowed indeterminate forms, the rule may be applied again when its conditions still hold.

For lim⁡x→0ex−1x,\lim_{x\to0}\frac{e^x-1}{x}, direct substitution gives 0/00/0, so L’Hospital’s Rule applies: lim⁡x→0ex1=1.\lim_{x\to0}\frac{e^x}{1}=1.

This is not the quotient rule: use f′/g′f'/g', not (f/g)′(f/g)'. Do not apply L’Hospital’s Rule unless the original limit is an allowed quotient form. Other indeterminate forms such as ∞−∞\infty-\infty are outside the assessed AP Calculus AB/BC scope stated here.