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Derivative-definition limit worksheets

Evaluate exact polynomial, reciprocal-power and trigonometric derivative limits at rational and pi-grid points.

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Find each limit exactly; do not replace radicals or fractions with decimals.

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Practice with exact answers

Keep fractions, radicals and pi exact. Round only when a question asks for a decimal answer, and check every original domain restriction.

Learning objective

What this activity builds

Recognize a supplied difference quotient as a derivative value and evaluate it exactly.

Worked example

How does a rate arise from a function?

Compare function values, then divide by the matching input change when the task asks for a rate.

Delta y, average rate, a derivative limit and an instantaneous rate are related but distinct quantities. Read the requested form before calculating.

Teaching notes

Make the practice count

  • Keep exact values through the final step.
  • Retain every original domain restriction.
  • Treat trigonometric inputs as radians.
  • Use the given seed to reproduce a worksheet.

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Math SheetLimits from the Derivative DefinitionFind each exact limit.1.lim h->0 [((3/2+h)^5)-((3/2)^5)]/h2.lim h->0 [((-2)/((4/3+h)^3))-((-2)/((4/3)^3))]/h3.lim h->0 [(-4sin(1(3/4pi+h)+0))-(-4sin(1(3/4pi)+0))]/h4.lim h->0 [(2cos(1(1/3pi+h)+0))-(2cos(1(1/3pi)+0))]/h5.lim h->0 [(3tan(1(2/3pi+h)+0))-(3tan(1(2/3pi)+0))]/h6.lim h->0 [(2(3/2+h)^5+1)-(2(3/2)^5+1)]/hmath-sheet.comPage 1 of 2
Math SheetLimits from the Derivative DefinitionFind each exact limit.7.lim h->0 [((-3)/((4/3+h)^3))-((-3)/((4/3)^3))]/h8.lim h->0 [(4sin(1(3/4pi+h)+0))-(4sin(1(3/4pi)+0))]/hmath-sheet.comPage 2 of 2

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