KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 05 · Lesson 08

Harmonic Comparisons and Model Limitations

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Harmonic Comparisons and Model Limitations

U05 · L08
1

Find the average rate of change of x² from 2 to 5.

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: 7
Reasoning: (25−4)/(5−2)=7.
2

Simplify [(x+h)²−x²]/h.

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: 2x+h
Reasoning: Expand and cancel h.
3

Evaluate lim_{x→3}(2x+1).

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: 7
Reasoning: Polynomials are continuous.
4

Evaluate lim_{x→2}(x²−4)/(x−2).

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: 4
Reasoning: Factor to x+2 for x≠2.
5

Can a two-sided limit exist if one-sided limits differ?

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: No
Reasoning: Both one-sided limits must agree.
6

Name the discontinuity in (x²−1)/(x−1) at x=1.

Limits visualPrepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. xy
Answer: Removable
Reasoning: The factor cancels but the original function is undefined there.
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