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Math Practice

Squeeze Theorem Practice Problems

Trap each function between two simpler ones with the same limit. Start from −1 ≤ sin ≤ 1 or −1 ≤ cos ≤ 1, multiply through, and use absolute values when the multiplier can be negative.

  1. 1.

    Find .

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    , and both bounds approach 0.

  2. 2.

    If for all , find .

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    Both bounds approach 5, so is squeezed to 5.

  3. 3.

    Find .

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    , and both bounds approach 0.

  4. 4.

    If for all , find .

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    Both bounds approach 1.

  5. 5.

    Find .

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    For , , and both bounds approach 0.

  6. 6.

    Find .

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    The factor changes sign, so bound the absolute value: . Both bounds approach 0.

  7. 7.

    If for , find .

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    The lower bound approaches and the upper bound .

  8. 8.

    If for all , find .

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    The lower bound approaches 2 and the upper bound .

  9. 9.

    Find .

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    , so . Both bounds approach 0.

  10. 10.

    Find .

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    , so . Both bounds approach 0.

  11. 11.

    Find .

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    , so . Both bounds approach 0 from the right.

  12. 12.

    Use (for small ) to find .

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    and the upper bound is 1, so the squeeze theorem gives 1.
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MPhil in Mathematics, Founder of Mathovia
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Squeeze Theorem Practice Problems

These problems follow the squeeze theorem lesson. If near and and share the limit , then has limit too.

Four steps every time

  1. Bound the wild part, usually with .
  2. Multiply through by the rest, using if the multiplier can be negative.
  3. Take the limit of each bound.
  4. If they agree, that common value is the limit.

Common mistakes to watch for

  • Bounds with different limits. The two bounds must approach the same value, or the theorem says nothing.
  • Multiplying by a negative without flipping. Bound the absolute value when the multiplier can change sign.
  • Using the product law first. does not exist, so it cannot be split off.
  • Forgetting where the inequality must hold. It needs to hold for near , not everywhere.

Frequently Asked Questions

Why are so many of the answers 0?+

Most squeeze theorem examples multiply a bounded, wiggling part by something that goes to 0, and bounded times zero is zero. The given-inequality problems are where other values appear.

Do I need to show the bounds?+

Only your final number is checked here, but on a test the bounds are the whole argument, so write them down each time.

More practice problems

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