Determine limits of functions resulting in indeterminate form.
I can recognize multiple indeterminate forms
I can apply L’Hospital’s rule when appropriate
Quick Lesson Plan
Many teachers present L’Hopital’s Rule without providing their students the opportunity to investigate the beauty of the concept behind the algorithm. We believe this lesson presents more content than simply applying derivative rules. The lesson reviews strategies for evaluating limits and then looks at the meaning and interpretation of the indeterminate forms that were discussed in Unit 1.
When evaluating the original limit expression, students will arrive at one of two indeterminate forms found in the AB curriculum (if you have time in the year --- or perhaps after the AP Test --- students will enjoy an extra day to investigate the other indeterminate forms that can be addressed by L’Hospital’s rule). To resolve this indeterminate form, we are directing our students to consider the rate at which the numerator and denominator approach zero or +/- infinity. This is the mechanism and the beauty of L’Hospital’s: to compare the rates of change of the original functions.
These problems appear in isolation on the MC section of the AB Calculus Test (until just a few years ago, the study of L’Hospital’s Rule was BC material) and usually as one part of a multi-step function analysis problem on the FRQ section. See 2013 BC 5a for an excellent example of L’Hospital’s Rule in use.
Careful! Don’t let your students forget to identify individually the limit results for the numerator and the denominator. Please look at our margin notes, Important Ideas, and solutions for the Check Your Understanding section to see proper notation for an indeterminate form. Also, students must know when to stop incorporating L’Hospital’s Rule: a real-number value in the numerator or denominator (other than 0 divided by 0) usually signals the end of the process.