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. Both one-sided limits at 6 equal -4 and f(6) = 9. Which listed change preserves the two-sided limit -4?

    • Change only f(6) from 9 to 0.
    • Change the left-hand limit to 0, keeping the right-hand limit at -4.
    • Change the right-hand limit to 0, keeping the left-hand limit at -4.
    • Change both one-sided limits from -4 to the new limiting value 0.

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

    • -1
    • 0
    • 1

  3. If both one-sided limits exist as finite numbers and agree, the finite two-sided limit equals their 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 supplied function has unequal finite one-sided limits at 7?

    • f(x) = x+2 on both sides
    • f(x) = 2 below 7 and 5 above 7
    • f(x) = x² on both sides
    • f(x) = 4 on both sides

  6. If the left side keeps oscillating without approaching any value or infinite direction, a right-hand limit of 3 does not establish a two-sided limit.

    Circle one:   True   False

  7. The left limit is 8 and the right limit is 2. Someone reports a limit of 5 by averaging. Why does that calculation not establish a limit?

    • Averaging always gives the larger approaching value.
    • A two-sided limit uses only the right-hand approaching value.
    • The assigned point value must equal the arithmetic mean.
    • Averaging does not make the approaching values agree.

  8. At 3 the left limit is -2 and the right limit is 7. Which assignment of f(3) makes the two-sided limit exist?

    • Set f(3) equal to the left limit -2.
    • Set f(3) equal to the right limit 7.
    • Set f(3) equal to their mean 2.5.
    • No assignment changes the unequal limits.

  9. For f(x) = 1/x at 0, why is there no two-sided infinite limit?

    • Both sides approach zero as x approaches zero.
    • The sides tend to opposite infinities.
    • The sides both approach positive infinity.

  10. For x distinct from 0, let h(x) = 1/x + 1/|x|. What happens as x approaches 0?

    • Both sides approach positive infinity.
    • Both sides approach the finite value zero.
    • The left approaches zero and the right positive infinity.
    • The left approaches negative infinity and the right zero.
LightMySky · lightmysky.comW1-mt__L0B_WQq-P-s1

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. Change only f(6) from 9 to 0. · Changing only f(6) leaves both one-sided limits at -4. Changing one limiting value destroys agreement; changing both to 0 changes the limit.
  2. 1 · Positive inputs give |x| = x, so the quotient is 1.
  3. True · Matching finite one-sided limits establish the finite two-sided limit.
  4. -1 · Negative inputs give |x| = -x, so the quotient is -1.
  5. f(x) = 2 below 7 and 5 above 7 · The constant branches 2 and 5 approach different finite values.
  6. True · The unsettled left side supplies no one-sided limit to agree with the right side.
  7. Averaging does not make the approaching values agree. · The arithmetic mean is 5, but the sides still approach different values.
  8. No assignment changes the unequal limits. · An assigned point value cannot change the unequal approaching values.
  9. The sides tend to opposite infinities. · The left tends to negative infinity and the right to positive infinity.
  10. The left approaches zero and the right positive infinity. · For negative x the terms cancel; for positive x they add to 2/x.
Worksheet · LightMySky