Limits that go to infinity at an asymptote

Answer Key - Grade 12 Math Worksheet

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

1. 1.

   Ari studies 2/(x − 3) as x → 3⁺. What is the limit?

   • f(x) → −∞ as x → 3⁺
   • f(x) → 0 as x → 3⁺
   • f(x) → +∞ as x → 3⁺
2. 2.

   Esme studies -3/(x − 1)³ as x → 1⁺. What is the limit?

   • f(x) → −∞ as x → 1⁺
   • f(x) → +∞ as x → 1⁺
   • f(x) → 0 as x → 1⁺
3. 3.

   Jun studies 4/(x − 5)³ as x → 5⁻. What is the limit?

   • f(x) → −∞ as x → 5⁻
   • f(x) → 0 as x → 5⁻
   • f(x) → +∞ as x → 5⁻
4. 4.

   Luz studies -2/(x + 1)² near x = -1, from both sides. What is the limit as x → -1?

   • no limit as x → -1
   • f(x) → +∞ as x → -1
   • f(x) → −∞ as x → -1
5. 5.

   Milo studies 4/(x − 3) near x = 3, from both sides. What is the limit as x → 3?

   • no limit as x → 3
   • f(x) → +∞ as x → 3
   • f(x) → −∞ as x → 3
6. 6.

   Noah studies -5/(x − 1) near x = 1, from both sides. What is the limit as x → 1?

   • no limit as x → 1
   • f(x) → −∞ as x → 1
   • f(x) → +∞ as x → 1
7. 7.

   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?

   4
8. 8.

   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?

   3
9. 9.

   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?

   -3
10. 10.

    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?

    -1