Limit laws and direct substitution

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   Ari substitutes first to find the limit of (x² − 4)/(x − 2) as x → 2. That gives a top of 0 and a bottom of 0. What should Ari conclude?

   • substitute: the limit is 0
   • k/0: no finite limit
   • 0/0: cancel (x − 2) first
2. 2.

   Esme substitutes first to find the limit of (x² − 9)/(x + 3) as x → -3. That gives a top of 0 and a bottom of 0. What should Esme conclude?

   • k/0: no finite limit
   • 0/0: cancel (x + 3) first
   • substitute: the limit is 0
3. 3.

   Jun substitutes first to find the limit of (x² + x)/(x − 2) as x → 2. That gives a top of 6 and a bottom of 0. What should Jun conclude?

   • substitute: the limit is 0
   • 0/0: rewrite it first
   • 6/0: no finite limit
4. 4.

   g(2) = 5; f(2) = 1; f(5) = 7. Kira wants the limit of f(g(x)) as x → 2, using the table shown. Both functions are smooth. What is the limit?

   7
5. 5.

   g(1) = 4; f(1) = 8; f(4) = 9. Luz wants the limit of f(g(x)) as x → 1, using the table shown. Both functions are smooth. What is the limit?

   9
6. 6.

   g(-1) = 3; f(-1) = -4; f(3) = 2. Milo wants the limit of f(g(x)) as x → -1, using the table shown. Both functions are smooth. What is the limit?

   2
7. 7.

   g(3) = -2; f(3) = 10; f(-2) = 5. Noah wants the limit of f(g(x)) as x → 3, using the table shown. Both functions are smooth. What is the limit?

   5
8. 8.

   Pia is told that the limit of 3x² + kx as x → 2 is 20. What is k?

   4
9. 9.

   Tala is told that the limit of 5x² + kx as x → -1 is 9. What is k?

   -4
10. 10.

    Zee is told that the limit of 4x² + kx as x → 2 is 6. What is k?

    -5