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Limits at Infinity Practice Problems

Find the dominant term. For rational functions compare degrees; for square roots remember √x² = |x|; for ∞ − ∞, multiply by the conjugate first.

  1. 1.

    Find .

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    A constant over something growing without bound approaches 0.

  2. 2.

    Find .

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    Equal degrees: the ratio of leading coefficients is .

  3. 3.

    Find .

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    The bottom has the higher degree, so the limit is 0.

  4. 4.

    Find .

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    Equal degrees: .

  5. 5.

    Find .

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    The top degree is higher; the expression behaves like , which grows without bound.

  6. 6.

    Find .

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    The expression behaves like , which goes to as .

  7. 7.

    Find .

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    Divide by : .

  8. 8.

    Find .

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    For , , which flips the sign: the limit is .

  9. 9.

    Find .

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    Conjugate: . Divide by : .

  10. 10.

    Find .

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    , and the bottom grows without bound.

  11. 11.

    Find , to three decimal places.

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    Inverse tangent levels off at .

  12. 12.

    Find .

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    Split: . The second term is squeezed between and , so it approaches 0.
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Limits at Infinity Practice Problems

These problems follow the limits at infinity lesson.

The degree rule for rational functions

Top degree vs bottom degreeLimit as
top smaller0
equalratio of leading coefficients
top larger or (check the sign)

Common mistakes to watch for

  • Treating as 1. It is indeterminate; compare degrees or divide by the highest power.
  • Dividing by the wrong power. Divide every term by the highest power of in the denominator.
  • Forgetting . As , pulling out of a square root flips the sign.
  • Assuming . Combine or use the conjugate before taking the limit.

Frequently Asked Questions

How do I enter an infinite answer?+

Type infinity or -infinity (∞ and −∞ also work). Use them when the function grows or falls without bound.

Can I type pi/2?+

Yes. Exact forms like pi/2 and decimals within 0.001 are both accepted.

More practice problems

Infinite Limits Practice Problems Evaluating Limits Practice Problems Squeeze Theorem Practice Problems