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L'Hôpital's Rule Practice Problems

Substitute first. If you get 0/0 or ∞/∞, differentiate the top and bottom separately and take the limit again. Rewrite other indeterminate forms as fractions before using the rule.

  1. 1.

    Find .

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    : differentiate top and bottom, .

  2. 2.

    Find using L’Hôpital’s rule.

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    : .

  3. 3.

    Find .

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    : .

  4. 4.

    Find .

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    : .

  5. 5.

    Find .

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    twice: , then .

  6. 6.

    Find .

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    twice: , then .

  7. 7.

    Find .

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    twice: , then .

  8. 8.

    Find .

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    Twice: , then .

  9. 9.

    Find .

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    Rewrite as (). The rule gives .

  10. 10.

    Find .

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    : .

  11. 11.

    Find , to three decimal places.

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    Take logs: , so the limit is .

  12. 12.

    Use L’Hôpital’s rule or substitution, whichever applies: .

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    Substitution gives . This is not indeterminate, so L’Hôpital’s rule must not be used (it would wrongly give 1).
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L’Hôpital’s Rule Practice Problems

These problems follow the L’Hôpital’s rule lesson.

Before every application

  1. Substitute. Only and qualify.
  2. Differentiate top and bottom separately, not with the quotient rule.
  3. Substitute again. If it’s still indeterminate, repeat.
  4. Other forms (, , exponent forms): rewrite as a fraction or take logs first.

Common mistakes to watch for

  • Using the quotient rule. Differentiate the top and bottom separately.
  • Not checking the form. The rule only applies to and .
  • Not re-checking after each step. Stop as soon as substitution gives a number.
  • Skipping the rewrite. Turn and into a fraction first.

Frequently Asked Questions

Can I enter fractions?+

Yes. Fractions like 1/2 and decimals within 0.001 are both accepted.

Why does one problem not use the rule?+

Because its form isn't indeterminate. Using L'Hôpital's rule on an ordinary limit gives a wrong answer, so always check the form first.

More practice problems

Evaluating Limits Practice Problems Limits at Infinity Practice Problems Squeeze Theorem Practice Problems