Limits at infinity and end behavior

Answer Key - Grade 12 Math Worksheet

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

1. 1.

   Ari compares the strongest terms of f(x) = (4x² − 1)/(2x³ + x). What happens as x → ∞?

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

   Bo compares the strongest terms of f(x) = (3x³ + 2)/(x² − 5). What happens as x → ∞?

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

   Esme compares the strongest terms of f(x) = (5x + 3)/(x² − 2). What happens as x → ∞?

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

   Kira looks at f(x) = (2x³ + 1)/(x² − 4) far to the left. What happens as x → −∞?

   • f(x) → +∞ as x → −∞
   • f(x) → 2 as x → −∞
   • f(x) → −∞ as x → −∞
5. 5.

   Milo looks at f(x) = (5x⁵ − 2)/(x⁴ + 1) far to the left. What happens as x → −∞?

   • f(x) → +∞ as x → −∞
   • f(x) → 5 as x → −∞
   • f(x) → −∞ as x → −∞
6. 6.

   Noah looks at f(x) = (-x³ + 7)/(2x² − 3) far to the left. What happens as x → −∞?

   • f(x) → −∞ as x → −∞
   • f(x) → +∞ as x → −∞
   • f(x) → -1/2 as x → −∞
7. 7.

   A tank's salt level is S(t) = 30t/(200 + 5t) after t minutes. Pia asks where it levels off. What is the limit as t → ∞, in g per liter?

   6
8. 8.

   A club's app gains members: M(t) = 500t/(2t + 20) after t minutes. Ren asks where it levels off. What is the limit as t → ∞, in members?

   250
9. 9.

   Sugar dissolves in a pitcher: C(t) = 12t/(50 + 4t) after t minutes. Tala asks where it levels off. What is the limit as t → ∞, in g per cup?

   3
10. 10.

    A sunflower grows: h(t) = 90t/(15 + 2t) after t minutes. Zee asks where it levels off. What is the limit as t → ∞, in cm?

    45