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Grade 12 · unit review

Review: Introductory Calculus: Limits and Continuity

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.

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Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Find lim as x→7 of (x²−49)/(x−7).
Need a hint?
  • A rational expression may simplify for every nearby input except its missing point.
  • Cancellation is valid only away from the zero denominator.
2. Near x=0, f(x)=7 for x<0 and f(x)=8 for x>0; f(0)=17. Find the two-sided limit at 0.
Need a hint?
  • A two-sided limit exists only when both one-sided limits exist and agree.
  • Do not average the two one-sided limits or substitute the isolated point’s value.
3. For x≠7, f(x)=(x²−49)/(x−7); define f(7)=k. What k makes f continuous at 7?
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.
4. Find lim as x→∞ of (7x+1)/(8x−2).
Need a hint?
  • Dividing numerator and denominator by x reveals reciprocal terms that approach zero as x grows without bound.
  • Infinity is an approach direction rather than a number to substitute.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Find lim as x→9 of (x²−81)/(x−9). Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. 2. 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. 3. 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. 4. Find lim as x→∞ of (9x+1)/(10x−2). Candidate answers: (A) -1 (B) 0 (C) 10/9 (D) 9/10.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 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. 2. 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. 3. 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. 4. 9/10. Divide numerator and denominator by x: (9+1/x)/(10−2/x). The reciprocal terms approach zero, leaving 9/10. The denominator approaches nonzero 10.

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