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Epsilon-Delta Practice Problems

For each limit, rewrite |f(x) − L| as a multiple of |x − a|, then choose δ so that multiple stays below ε. For quadratics, cap δ at 1 first to bound the extra factor.

  1. 1.

    For , . What is ?

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    .

  2. 2.

    For and , what is the largest that works?

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    when .

  3. 3.

    For and , what is the largest that works?

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    , so .

  4. 4.

    For and , what is the largest that works?

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    , so .

  5. 5.

    For and , what is the largest that works?

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    , so .

  6. 6.

    For and , what is the largest that works?

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    , so .

  7. 7.

    For a line with , the rule gives . What is ?

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    , so .

  8. 8.

    For (a constant function), is acceptable for every ? Type yes or no.

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    for every , so any works.

  9. 9.

    For , if , what is the smallest whole number with guaranteed?

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    puts in , so is in and .

  10. 10.

    Using for , what is when ?

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    .

  11. 11.

    Using the same rule, what is when ?

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    . The cap of 1 keeps the bound valid.

  12. 12.

    For with , you get . With , find for .

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    .
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Epsilon-Delta Practice Problems

These problems follow the epsilon-delta definition of a limit lesson.

The method

  1. Simplify and write it as (or bound it that way).
  2. Make by taking .
  3. For a nonlinear limit, cap at 1 first to bound the extra factor, then take the minimum.

Common mistakes to watch for

  • Choosing from . may depend on only.
  • Wrong multiple. If , then .
  • Forgetting the cap for nonlinear limits. Restrict first, then take the minimum.
  • Picking yourself. is given; your job is to find .

Frequently Asked Questions

Is there only one correct δ?+

No. Any smaller positive δ also works. These problems ask for the largest δ that works, which is the one the standard method produces.

Why use min(1, ε/7)?+

Capping δ at 1 keeps x close enough to 3 that |x + 3| < 7. Then ε/7 makes 7|x − 3| less than ε. The minimum satisfies both conditions at once.

More practice problems

Introduction to Limits Practice Problems Limit Laws Practice Problems Absolute Value Inequalities Practice Problems