Require Left and Right Approaches to Agree

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Near x=0, f(x)=2 for x<0 and f(x)=3 for x>0; f(0)=12. Find the two-sided limit at 0. Candidate answers: (A) The limit is 12 (B) The limit is 2 (C) The limit is 3 (D) The two-sided limit does not exist.
• Independent check inputs: Near x=0, f(x)=3 for x<0 and f(x)=4 for x>0; f(0)=13. Find the two-sided limit at 0. Candidate answers: (A) The limit is 13 (B) The limit is 3 (C) The limit is 4 (D) The two-sided limit does not exist.

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. A two-sided limit exists only when both one-sided limits exist and agree. A function’s separately assigned value at the point cannot reconcile different left and right approach values.

Here is the method on the supplied teaching case. The left-hand limit is 2; the right-hand limit is 3. They differ, so no two-sided limit exists. Changing f(0) would not reconcile those approach values.

Do not average the two one-sided limits or substitute the isolated point’s value. Compare the approach values directly; different values imply that the two-sided limit does not exist.

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.

Check your reasoning

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Take it further

Changed-input task: Near x=0, f(x)=9 for x<0 and f(x)=10 for x>0; f(0)=19. Find the two-sided limit at 0. Candidate answers: (A) The limit is 10 (B) The limit is 19 (C) The limit is 9 (D) The two-sided limit does not exist. Show the mathematical reason, not only the selected answer. Expected result: The two-sided limit does not exist. The left-hand limit is 9; the right-hand limit is 10. They differ, so no two-sided limit exists. Changing f(0) would not reconcile those approach values.

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