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Math Practice

Evaluating Limits Practice Problems

Substitute first. When you get 0/0, factor, rationalize, combine fractions or expand until the zero factor cancels, then substitute again. A nonzero number over 0 means the limit does not exist.

  1. 1.

    Find .

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    Factor: .

  2. 2.

    Find .

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    , so the expression is .

  3. 3.

    Find .

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    Factor: .

  4. 4.

    Find .

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    Difference of cubes: , so the limit is .

  5. 5.

    Find .

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    , so the expression is .

  6. 6.

    Find .

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    Multiply by : .

  7. 7.

    Find .

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    Multiply by : the top becomes , leaving .

  8. 8.

    Find .

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    Expand: . Dividing by gives .

  9. 9.

    Find .

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    Combine: . Dividing by gives .

  10. 10.

    Find (radians).

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    Rewrite as , and .

  11. 11.

    Find (radians).

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    Write it as . Both fractions approach 1, leaving .

  12. 12.

    Find , or type DNE.

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    The top approaches 5 and the bottom approaches 0, so this is , not . The function blows up in opposite directions on each side, so the limit does not exist.
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Evaluating Limits Practice Problems

These problems follow the evaluating limits lesson. Substitution gives in almost all of them, so each needs one algebraic step before you can substitute.

Which technique?

The expression has…Try
polynomials on top and bottomfactor and cancel
a square rootmultiply by the conjugate
fractions inside a fractioncombine over a common denominator
a power like expand and cancel
near 0match the form

Common mistakes to watch for

  • Cancelling terms instead of factors. Factor first; only common factors cancel.
  • Forgetting the conjugate. For a square root in a limit, multiply by the conjugate.
  • Expanding the conjugate on the bottom. Leave it factored so the zero factor can cancel.
  • Degrees in trig limits. holds only in radians.

Frequently Asked Questions

Can I give a fraction as my answer?+

Yes. Exact fractions like 1/6 and decimals within 0.001 are both accepted.

How do I know which technique to use?+

Look at the expression: polynomials factor, square roots need the conjugate, fractions inside a fraction get a common denominator, and powers like (a + h)² get expanded.

More practice problems

Limit Laws Practice Problems Factoring Polynomials Practice Problems Rationalizing Denominators Practice Problems