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One-Sided Limits Practice Problems

For each limit, use only the values of x on the side the arrow comes from. Type DNE when a two-sided limit fails, and infinity or -infinity when the values grow without bound.

  1. 1.

    For when and when , find .

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    From the left use : .

  2. 2.

    For the same function ( for , for ), find .

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    From the right use : .

  3. 3.

    For the same function, does exist? Give its value, or type DNE.

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    The left-hand limit is 5 and the right-hand limit is 7. They disagree, so the two-sided limit does not exist.

  4. 4.

    Find .

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    For , , so the fraction equals 1.

  5. 5.

    For when and when , find , or type DNE.

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    Left: . Right: . Both sides agree, so the limit is 1.

  6. 6.

    Find .

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    For , is negative, so and the fraction is .

  7. 7.

    Find .

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    Just above 9, is a tiny positive number, and its square root approaches 0.

  8. 8.

    Find , where rounds down to the nearest integer.

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    Values just below 4, like 3.99, round down to 3.

  9. 9.

    Find . Type infinity or -infinity if it grows without bound.

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    At the denominator is and the fraction is 100. The values grow without bound.

  10. 10.

    Find . Type infinity or -infinity if it grows without bound.

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    At the denominator is and the fraction is . The values fall without bound.

  11. 11.

    For when and when , what value of makes exist?

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    The left-hand limit is . The right-hand limit is . Setting gives .

  12. 12.

    Find .

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    For , , so , which approaches .
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One-Sided Limits Practice Problems

These problems follow the one-sided limits lesson.

Quick method

  1. Pick the side. uses ; uses .
  2. Use the rule for that side, then substitute or simplify.
  3. For infinite limits, check the sign of the denominator just beside .
  4. For a two-sided limit, both sides must agree.

Common mistakes to watch for

  • Reading the minus as a negative number. approaches positive 2 from below.
  • Using the wrong piece. For , use the piece for .
  • Assuming one side is enough. The two-sided limit needs both sides to agree.
  • Guessing the sign of an infinite limit. Test a value just beside .

Frequently Asked Questions

How do I enter an infinite one-sided limit?+

Type infinity or -infinity (∞ and −∞ also work). Use it when the values grow or fall without bound as x approaches from that side.

Which piece of a piecewise function do I use?+

For x → a⁻ use the piece for x < a; for x → a⁺ use the piece for x > a. The piece that includes a itself only matters for the value f(a).

More practice problems

Finding Limits From a Graph Practice Problems Introduction to Limits Practice Problems Piecewise Functions Practice Problems