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Grade 12 · lesson 3 of 4

Choose a Value that Matches the Limit

Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For x≠2, f(x)=(x²−4)/(x−2); define f(2)=k. What k makes f continuous at 2? Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
  • Independent check inputs: For x≠3, f(x)=(x²−9)/(x−3); define f(3)=k. What k makes f continuous at 3? Candidate answers: (A) 12 (B) 5 (C) 6 (D) 7.

Learn the idea

A limit describes nearby approach behavior rather than necessarily a function’s value at the point. One-sided agreement, removable holes and continuity connect algebraic simplification with precise statements about a function’s local behavior. Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit. A removable hole can be filled with its approach value to meet these conditions.

Here is the method on the supplied teaching case. The nearby simplified rule is x+2, whose limit at 2 is 4. Continuity requires the defined function value to equal that limit, so k=4.

Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits. State all three continuity conditions and retain the distinction between the original rule and the newly defined extension.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: For x≠2, f(x)=(x²−4)/(x−2); define f(2)=k. What k makes f continuous at 2? Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: The nearby simplified rule is x+2, whose limit at 2 is 4. Continuity requires the defined function value to equal that limit, so k=4. The final answer is 4. Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits. State all three continuity conditions and retain the distinction between the original rule and the newly defined extension.

Your turn

Check your reasoning

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Helpful tips
  • Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.
  • Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits.
For x≠3, f(x)=(x²−9)/(x−3); define f(3)=k. What k makes f continuous at 3?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

6

The nearby simplified rule is x+3, whose limit at 3 is 6. Continuity requires the defined function value to equal that limit, so k=6.

Take it further

Changed-input task: For x≠9, f(x)=(x²−81)/(x−9); define f(9)=k. What k makes f continuous at 9? Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. Show the mathematical reason, not only the selected answer. Expected result: 18. The nearby simplified rule is x+9, whose limit at 9 is 18. Continuity requires the defined function value to equal that limit, so k=18.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For x≠4, f(x)=(x²−16)/(x−4); define f(4)=k. What k makes f continuous at 4?
Need a hint?
  • Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.
  • Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits.
2. For x≠5, f(x)=(x²−25)/(x−5); define f(5)=k. What k makes f continuous at 5?
Need a hint?
  • Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.
  • Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits.
3. For x≠6, f(x)=(x²−36)/(x−6); define f(6)=k. What k makes f continuous at 6?
Need a hint?
  • Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.
  • Changing a point’s value can fix a removable hole but cannot fix unequal one-sided limits.

Your choices are temporary. Feedback appears after you check all the answers.

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