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Limits at Essential Discontinuities worksheets

Evaluate exact one-sided and two-sided limits at rational and trigonometric poles.

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Evaluate each requested limit without sampling numerically.

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Approach from each side

Every task keeps the original equation, domain and point value separate from its exact one-sided behavior.

Learning objective

What this activity builds

Classify signed unbounded behavior at finite poles exactly.

Worked example

At x = a, the left branch approaches 2 and the right branch approaches -1. Find all three limits.

The one-sided limits are 2 and -1; the two-sided limit is DNE.

Unequal one-sided limits prevent a two-sided limit, regardless of f(a).

Teaching notes

Make the practice count

  • Read the approach direction before evaluating.
  • Keep signed infinity separate from finite real limits.
  • Check open and closed graph endpoints independently.

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Exact one-sided behavior drives both the equation and graph presentations.

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Math SheetNameScoreTeacherDateLimits at Essential DiscontinuitiesEvaluate each requested finite-point limit exactly.1.f(x) =2x+2·x+2·x+2limx -> -2+f(x) = ________________2.f(x) =2-cot-1·x+2limx -> -2-f(x) = ________________3.f(x) =12x-1limx -> 1-f(x) = ________________4.f(x) =-1·x·x-1·x-2x+1limx -> -1-f(x) = ________________Keep one-sided limits, two-sided limits, and f(a) distinct.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateLimits at Essential DiscontinuitiesEvaluate each requested finite-point limit exactly.5.f(x) =-2-2·csc12·x+1limx -> -1+f(x) = ________________6.f(x) =12x+2·x+2limx -> -2+f(x) = ________________7.f(x) =-2·x+1·x-1x-2·x-2·x-2limx -> 2+f(x) = ________________8.f(x) =0-2·tan12·+x+2limx -> -2-f(x) = ________________Keep one-sided limits, two-sided limits, and f(a) distinct.math-sheet.comPage 2 of 2

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