Florida B.E.S.T. MA.912.C.1.9

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

The Standard

Define continuity in terms of limits.

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

Teacher's field guide

What Students Need to Do: Students decide whether a function is continuous at a given input. They check the function value, the left and right limits, and whether the shared limit matches the function value.
What Mastery Looks Like: A student checks that f(a) exists, the two-sided limit exists, and the limit equals f(a). The student can identify which condition fails at a hole, jump, or asymptote.
Common Misconceptions: Students may think a graph is continuous whenever a limit exists. They may ignore an undefined point, confuse one-sided limits, or assume the function value determines the limit.
How to Assess It: Give students f(x) = (x² - 4)/(x - 2) for x ≠ 2 and f(2) = 5. Ask whether f is continuous at x = 2 and require all three conditions.

Lesson moves

Give pairs graph cards showing holes, jumps, asymptotes, and smooth curves, then have them sort the cards by which continuity condition fails.
Ask students to explain why filling a hole can make a function continuous, but changing one point cannot repair a jump.
Play a three-condition check game where students inspect functions and earn one point for correctly naming the first failed condition.
Graph a taxi fare with a fixed starting fee, then discuss where jumps occur and what those jumps mean in the pricing rule.

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