1.Evaluate the limit analytically: lim (x -> 3) [(x² - 9) / (x - 3)]
A0
BUndefined (Division by zero)
C6
D3
Working
Grade 12algebraOriginal practice
Calculus: Limits, Continuity & Derivative Rules
Evaluate limits algebraically, compute instantaneous rates of change, and apply the Power Rule to determine derivative functions and tangent slopes.
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About this practice
Grade 12 · math · Matching answer key included · US Letter and A4
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Teaching guide, sample answers and curriculum references
Before you begin
Simplify algebraic expressions and understand a function's input and output.
How to use this worksheet
- Model the target skill: evaluate two-sided limits analytically using direct substitution, factoring, and algebraic rationalization. Use the first task as a guided example before asking for an independent response.
- Check readiness first: Simplify algebraic expressions and understand a function's input and output.
- Discuss an incorrect response using this distinction: A function value and a limit are different; direct substitution may fail even when a limit exists.
A common mistake to discuss
A function value and a limit are different; direct substitution may fail even when a limit exists.
What to practice next
Compare the limiting behavior and the function value at one removable discontinuity.
Sample question and answer
Evaluate the limit analytically: lim (x -> 3) [(x² - 9) / (x - 3)]
- A. 0
- B. Undefined (Division by zero)
- C. 6
- D. 3
Answer: C. Factoring the numerator yields (x - 3)(x + 3) / (x - 3). For x ≠ 3, cancel (x - 3): lim (x -> 3) (x + 3) = 3 + 3 = 6.
Intended curriculum references: CCSS.MATH.CONTENT.HSF.IF.B.4 · CCSS.MATH.CONTENT.HSF.IF.C.7. These references describe learning goals; they are not endorsements.
Published 2026-10-02; updated 2026-10-02. Original MakeWorksheet practice. Automated consistency checks support publication; independent subject-specialist review is not recorded. Read our editorial policy.
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