One-Sided Limits and When a Limit Fails to Exist · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Both sides must agree

Mathematics · Calculus & Analysis · ages 18-19
Name ______________________   Date ____________

  1. A graph is made of two pieces. As x approaches 2 from the left, the curve climbs toward height 3. As x approaches 2 from the right, the curve also climbs toward height 3. The filled dot at x = 2 sits at height 7. What is the limit of f(x) as x approaches 2?

    • 3
    • 7
    • The limit does not exist
    • 10

  2. What is the limit of |x| / x as x approaches 0 from the right?

    • -1
    • 0
    • 1

  3. If the left-hand and right-hand limits agree, the two-sided limit exists and equals that shared value.

    Circle one:   True   False

  4. What is the limit of |x| / x as x approaches 0 from the left?

    • -1
    • 0
    • 1

  5. Which function fails to have a limit at x = 0 because its one-sided values disagree?

    • f(x) = x squared
    • f(x) = |x| / x
    • f(x) = x + 1
    • f(x) = 3

  6. Saying a limit is infinite means the outputs grow without bound in one direction that both sides share. That is a different description from saying the limit fails to exist because the two sides disagree.

    Circle one:   True   False

  7. A piecewise graph jumps at x = 1. Coming in from the left, the curve settles at height 4. Coming in from the right, the curve settles at height 2. What is the limit of f(x) as x approaches 1?

    • 4
    • 2
    • 3
    • The limit does not exist

  8. A piecewise function is f(x) = 2x + 1 when x is less than 2, and f(x) = 8 - x when x is greater than 2. What is the limit of f(x) as x approaches 2?

    • 5
    • 6
    • 5.5
    • The limit does not exist

  9. For f(x) = 1 / x at 0, a friend calls the limit infinite. Why is that label wrong?

    • The limit equals 0 instead
    • The two sides race in opposite directions, so the limit does not exist
    • Infinite limits only happen at jumps

  10. For f(x) = 1 / x as x approaches 0, the left side drops toward negative infinity and the right side grows toward positive infinity. How should the two-sided limit be described?

    • It equals 0
    • It is infinite
    • It does not exist, because the two sides disagree
    • It equals 1
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Both sides must agree W1-mt__L0B_WQq-P-s1

  1. 3 · A limit only cares about where the outputs are heading, not where the dot actually is. Both sides head to 3, so the limit is 3. The dot at 7 affects continuity, not the limit.
  2. 1 · From the right x is positive, and there the fraction equals 1.
  3. True · Agreement is the whole test for a two-sided limit.
  4. -1 · From the left x is negative, and there the fraction equals -1.
  5. f(x) = |x| / x · For |x| / x, the right side gives 1 and the left side gives -1, so the one-sided limits disagree and the limit fails. The other options settle on one value from both sides.
  6. True · Both ideas involve failure to settle on a number, but they say different things. Infinite means unbounded in a shared direction. Failing to exist means the one-sided targets, whether numbers or directions, do not match.
  7. The limit does not exist · The left-hand limit is 4 and the right-hand limit is 2. Since the two sides disagree, there is no single target, so the limit does not exist. Averaging to 3 is a common mistake.
  8. The limit does not exist · The left piece gives 2(2) + 1 = 5 and the right piece gives 8 - 2 = 6. Since 5 and 6 disagree, the limit does not exist.
  9. The two sides race in opposite directions, so the limit does not exist · Infinite needs both sides to agree on one unbounded direction.
  10. It does not exist, because the two sides disagree · Here one side is unbounded up and the other unbounded down. The two one-sided behaviors never agree, so the two-sided limit does not exist. An infinite limit needs both sides to agree on the unbounded direction.
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