Grade 12 · Math · Calculus

12th grade Limits that go to infinity at an asymptote worksheet

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Limits that go to infinity at an asymptote

Grade 12 Math Worksheet

NameDate

Circle the best answer, or write your answer on the line.

1.
Ari studies 2/(x − 3) as x → 3⁺. What is the limit?
- f(x) → −∞ as x → 3⁺
- f(x) → +∞ as x → 3⁺
- f(x) → 0 as x → 3⁺
2.
Bo studies 5/(x + 2) as x → -2⁻. What is the limit?
- f(x) → 0 as x → -2⁻
- f(x) → −∞ as x → -2⁻
- f(x) → +∞ as x → -2⁻
3.
Esme studies -3/(x − 1)³ as x → 1⁺. What is the limit?
- f(x) → 0 as x → 1⁺
- f(x) → +∞ as x → 1⁺
- f(x) → −∞ as x → 1⁺
4.
Jun studies 4/(x − 5)³ as x → 5⁻. What is the limit?
- f(x) → 0 as x → 5⁻
- f(x) → −∞ as x → 5⁻
- f(x) → +∞ as x → 5⁻
5.
Milo studies 4/(x − 3) near x = 3, from both sides. What is the limit as x → 3?
- f(x) → −∞ as x → 3
- no limit as x → 3
- f(x) → +∞ as x → 3
6.
Noah studies -5/(x − 1) near x = 1, from both sides. What is the limit as x → 1?
- f(x) → −∞ as x → 1
- f(x) → +∞ as x → 1
- no limit as x → 1
Pia graphs f(x) = (x − 1)/((x − 1)(x − 4)). One zero of the bottom is a hole, not an asymptote. Where is the vertical asymptote?
Ren graphs f(x) = (x + 2)/((x + 2)(x − 3)). One zero of the bottom is a hole, not an asymptote. Where is the vertical asymptote?
Tala graphs f(x) = (x − 5)/((x − 5)(x + 3)). One zero of the bottom is a hole, not an asymptote. Where is the vertical asymptote?
Zee graphs f(x) = (x + 4)/((x + 4)(x + 1)). One zero of the bottom is a hole, not an asymptote. Where is the vertical asymptote?

Limits that go to infinity at an asymptote. Every version has 10 questions, and the answer key prints on the second page.

I can find a vertical asymptote and tell a hole from an asymptote.
I can decide whether f goes to +∞ or −∞ on each side of an asymptote.
I can say when the two-sided limit at an asymptote exists as ∞ or −∞, and when it does not exist.

Standard: none (beyond CCSS)

vertical asymptote: A vertical line x = a the graph runs up or down beside.

+∞, −∞: The values grow without bound, upward or downward.

hole: A single missing point where a factor cancelled.

Used the zero of the top. The top's zero is where the graph crosses the axis; the bottom's zero gives the asymptote.

Read x − 3 = 0 as x = −3. Put the answer back in: the factor should be 0.

Read the curve on the other side of the asymptote. From the left means x a little less than the asymptote's x.

What is the most common mistake with limits that go to infinity at an asymptote?

Used the zero of the top. The top's zero is where the graph crosses the axis; the bottom's zero gives the asymptote.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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