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Introduction to Limits Practice Problems

Each problem asks what value f(x) approaches. Substitute when nothing breaks, simplify when you get 0/0, and type DNE when the two sides disagree or the values never settle.

  1. 1.

    Find .

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    Nothing breaks at , so substitute: .

  2. 2.

    Find .

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    Substitute: .

  3. 3.

    A function has for and . Find .

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    Near 2 the function follows , which approaches 7. The single value doesn’t affect the limit.

  4. 4.

    Find .

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    For the expression equals , which approaches 2.

  5. 5.

    Find .

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    for , which approaches 6.

  6. 6.

    Find .

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    , so the expression is for , which approaches 1.

  7. 7.

    Use a table of values to estimate (radians).

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    gives 0.99998, and negative inputs give the same values. They close in on 1.

  8. 8.

    Use a table of values to estimate .

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    gives about 1.005 and about 0.995. Both sides approach 1.

  9. 9.

    Does exist? Give its value, or type DNE.

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    The expression is 1 for and for . The two sides disagree, so the limit does not exist.

  10. 10.

    Does exist? Give its value, or type DNE.

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    The values grow without bound on both sides ( gives 10,000). Infinity isn’t a number, so the limit does not exist.

  11. 11.

    A ball has fallen metres after seconds. Find , its speed in m/s at .

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    , which approaches 20.

  12. 12.

    Find .

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    Expand: for , which approaches 6.
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Introduction to Limits Practice Problems

These problems follow the introduction to limits lesson. Each one asks for , the value approaches as gets close to .

Three things to try

  1. Substitute. If you get a number, that’s the limit for everyday functions.
  2. If you get , simplify. Factor and cancel, then substitute again.
  3. Check both sides. If they head for different values, or off to infinity, the limit does not exist.

Common mistakes to watch for

  • Treating the limit as . They match for continuous functions, but the limit can exist where does not.
  • Stopping at . It means more work is needed: simplify, then substitute again.
  • Checking only one side. Both sides must approach the same value.
  • Trusting a table as proof. A table suggests a limit; algebra confirms it.

Frequently Asked Questions

How do I enter a limit that does not exist?+

Type DNE (or "does not exist"). Use it when the left and right sides approach different values, or when the values grow without bound.

Is 0/0 an answer?+

No. 0/0 means substitution failed and more work is needed, such as factoring and cancelling, or estimating with a table.

More practice problems

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