Florida B.E.S.T. MA.912.C.1.8

MathGrades 9–12Develop an understanding for limits and continuity. Determine limits and continuity.

The Standard

Find limits of indeterminate forms using L'Hôpital's Rule.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students test a quotient by direct substitution and identify the forms 0/0 and ∞/∞. They differentiate the numerator and denominator separately, then reevaluate the limit, repeating when needed. They know when another method is required.
What Mastery Looks Like: A student writes the indeterminate form before applying the rule. They differentiate accurately, simplify, and find the limit after one or more rounds. They reject the rule when the required form is absent.
Common Misconceptions: Students often apply the rule to any quotient, including forms such as 1/0 or 0/∞. Some use the quotient rule instead of differentiating the top and bottom separately. Others stop when the new quotient is still indeterminate or assume ∞/∞ equals 1.
How to Assess It: Exit ticket: Evaluate lim x→0 of (e^x − 1 − x)/x². Show the substitution result, each derivative step, and the final value.

Lesson moves

Sort limit cards into 0/0, ∞/∞, and not eligible piles, then solve one card from each eligible pile.
Compare differentiating numerator and denominator separately with using the quotient rule, then write why only the first process fits here.
Run a three-round relay with limits needing zero, one, or two applications, and require teams to label each resulting form.
Use P(t)=1000e^(0.04t) to find lim t→0 of [P(t)−P(0)]/t and interpret the initial population growth rate.

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