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Infinite Limits Practice Problems

Substitute first: a nonzero number over 0 means the limit is infinite on each side. Then check the sign just left and just right of a to choose between infinity and -infinity.

  1. 1.

    Determine .

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    Positive over a positive square on both sides, so the values grow without bound.

  2. 2.

    Determine .

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    A negative numerator over a positive square on both sides.

  3. 3.

    Determine .

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    Just right of 5 the denominator is a tiny positive number.

  4. 4.

    Determine .

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    Just left of 5 the denominator is a tiny negative number.

  5. 5.

    Determine .

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    The top approaches 4 (positive); the bottom is a tiny negative number.

  6. 6.

    Determine , or type DNE if the sides disagree.

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    The left side goes to and the right side to , so there is no single infinite limit.

  7. 7.

    Determine .

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    For small negative , is a tiny negative number, so the fraction is hugely negative.

  8. 8.

    Determine .

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    The top approaches (negative); the bottom is a tiny positive number.

  9. 9.

    Determine .

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    . Just left of 2, is negative and , so the bottom is negative while the top is about 4.

  10. 10.

    Find . Type a number, infinity, -infinity or DNE.

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    This is , not an infinite limit. Cancel: . The graph has a hole at .

  11. 11.

    Determine .

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    As shrinks towards 0 from the right, falls without bound: , .

  12. 12.

    Determine .

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    Just left of , and is a tiny positive number, so grows without bound.
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MPhil in Mathematics, Founder of Mathovia
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Infinite Limits Practice Problems

These problems follow the infinite limits lesson.

How to determine an infinite limit

  1. Substitute. A nonzero number over 0 means the limit is infinite from each side; means simplify first.
  2. Check the sign just left of : positive gives , negative gives .
  3. Check the sign just right of the same way.
  4. Compare the sides. If they agree, that’s the two-sided limit; if not, the two-sided limit is neither.

Common mistakes to watch for

  • Saying the limit exists because it is . An infinite limit describes how the limit fails to exist.
  • Writing when the sides disagree. goes to on the left and on the right.
  • Not simplifying . A cancelled factor gives a hole with a finite limit, not an asymptote.
  • Guessing the sign. Test a value just beside to see whether the function heads up or down.

Frequently Asked Questions

How do I type an infinite limit?+

Type infinity or -infinity (∞ and −∞ also work). Type DNE when the two sides go to different infinities.

Why is one answer a number?+

One problem gives 0/0, which is not an infinite limit. The common factor cancels, leaving a hole with a finite limit.

More practice problems

One-Sided Limits Practice Problems Evaluating Limits Practice Problems Squeeze Theorem Practice Problems