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Types of Discontinuity Practice Problems

Find the one-sided limits at each point. Equal finite limits mean removable, different finite limits mean a jump, an infinite side means infinite, and endless wiggling means oscillating.

  1. 1.

    Classify the discontinuity of at .

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    It simplifies to , so the limit is 10, but is undefined: a hole.

  2. 2.

    Classify the discontinuity of at .

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    : both one-sided limits are infinite, so there is a vertical asymptote.

  3. 3.

    For when and when , classify the discontinuity at .

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    The left limit is 0 and the right limit is 2: finite but different.

  4. 4.

    Classify the discontinuity of at .

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    The values swing between and 1 faster and faster, so neither one-sided limit exists.

  5. 5.

    Classify the discontinuity of at .

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    The expression is on the left and 1 on the right: a jump of size 2.

  6. 6.

    Classify the discontinuity of at .

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    The top approaches 4 and the bottom approaches 0 through positive values, so both sides go to .

  7. 7.

    What value of removes the discontinuity of ?

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    .

  8. 8.

    A function has and . What is the size of the jump at ?

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    The jump is the difference between the one-sided limits: .

  9. 9.

    . At which does have a removable discontinuity?

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    : the factor cancels, leaving a hole at .

  10. 10.

    For the same function , at which is the discontinuity infinite?

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    After cancelling, still has in the denominator, which is zero at .

  11. 11.

    For , what is the -coordinate of the hole? Give a decimal or a fraction.

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    The simplified form at is .

  12. 12.

    for and . Classify the discontinuity at .

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    The limit is 4 and exists, but is a different number: a moved point, which is removable.
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MPhil in Mathematics, Founder of Mathovia
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Types of Discontinuity Practice Problems

These problems follow the types of discontinuity lesson.

One-sided limits at Type
finite and equal, but missing or differentremovable
finite but differentjump
at least one is infinite
at least one never settlesoscillating

Common mistakes to watch for

  • Calling every zero of the denominator an asymptote. Cancel first; a cancelled factor leaves a hole.
  • Judging a jump from one side. You need both one-sided limits.
  • Thinking a dot fixes a jump. A filled dot does not help if the two sides approach different heights.
  • Forgetting piecewise switch points. They are the most common place for a jump.

Frequently Asked Questions

How do I type my answer?+

Type one word: removable, jump, infinite or oscillating. For the number questions, type the value.

Is a moved point removable?+

Yes. If the limit exists but f(a) is a different number, redefining f(a) as the limit removes the discontinuity.

More practice problems

Continuity Practice Problems Infinite Limits Practice Problems One-Sided Limits Practice Problems