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Limits at Jump Discontinuities worksheets

Compare exact left-hand, right-hand and two-sided limits at piecewise junctions.

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Find the left-hand, right-hand and two-sided limit at every junction.

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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

Separate one-sided limits, two-sided existence and function values.

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).

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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 Jump DiscontinuitiesFind the left-hand, right-hand and two-sided limit at each junction.1.f(x) =-4, x < 0-2, x > 0limx -> 0-f(x) = ________________limx -> 0+f(x) = ________________limx -> 0f(x) = ________________2.f(x) = -1 + 2|x + 1|x + 1limx -> -1-f(x) = ________________limx -> -1+f(x) = ________________limx -> -1f(x) = ________________3.f(x) =x-52, x < 1x-12, x > 1f(1) = -3/2limx -> 1-f(x) = ________________limx -> 1+f(x) = ________________limx -> 1f(x) = ________________4.f(x) = 2 + 2|x - 1|x - 1limx -> 1-f(x) = ________________limx -> 1+f(x) = ________________limx -> 1f(x) = ________________Keep one-sided limits, two-sided limits, and f(a) distinct.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateLimits at Jump DiscontinuitiesFind the left-hand, right-hand and two-sided limit at each junction.5.f(x) =12·x2+2·x+32, x < -112·x2+2·x+12, x > -1f(-1) = -1limx -> -1-f(x) = ________________limx -> -1+f(x) = ________________limx -> -1f(x) = ________________6.f(x) = -2 - 2|x - 1|x - 1limx -> 1-f(x) = ________________limx -> 1+f(x) = ________________limx -> 1f(x) = ________________7.f(x) =2, x < 10, x > 1f(1) = 4limx -> 1-f(x) = ________________limx -> 1+f(x) = ________________limx -> 1f(x) = ________________8.f(x) = 2|x - 2|x - 2limx -> 2-f(x) = ________________limx -> 2+f(x) = ________________limx -> 2f(x) = ________________Keep one-sided limits, two-sided limits, and f(a) distinct.math-sheet.comPage 2 of 2

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