Find an Approach Value across a Removable Hole
A rational expression may simplify for every nearby input except its missing point.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Find lim as x→2 of (x²−4)/(x−2). Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
- Independent check inputs: Find lim as x→3 of (x²−9)/(x−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. A rational expression may simplify for every nearby input except its missing point. The simpler rule then describes approach behavior, allowing a limit even when the original function is undefined there.
Here is the method on the supplied teaching case. For x≠2, factor and cancel to get x+2. As x approaches 2, this approaches 4. The original function is still undefined exactly at the excluded point.
Cancellation is valid only away from the zero denominator. A computed limit does not fill the original domain hole by itself; write the distinction between nearby values and the value at the point.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Find lim as x→2 of (x²−4)/(x−2). Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: For x≠2, factor and cancel to get x+2. As x approaches 2, this approaches 4. The original function is still undefined exactly at the excluded point. The final answer is 4. Cancellation is valid only away from the zero denominator. A computed limit does not fill the original domain hole by itself; write the distinction between nearby values and the value at the point.
Check your reasoning
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Helpful tips
- A rational expression may simplify for every nearby input except its missing point.
- Cancellation is valid only away from the zero denominator.
See the answer and explanation
6
For x≠3, factor and cancel to get x+3. As x approaches 3, this approaches 6. The original function is still undefined exactly at the excluded point.
Take it further
Changed-input task: Find lim as x→9 of (x²−81)/(x−9). Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. Show the mathematical reason, not only the selected answer. Expected result: 18. For x≠9, factor and cancel to get x+9. As x approaches 9, this approaches 18. The original function is still undefined exactly at the excluded point.
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