Grade 12 · Math · Calculus

12th grade Limit laws and direct substitution worksheet

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Limit laws and direct substitution

Grade 12 Math Worksheet - Challenge Me

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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.
Bo substitutes first to find the limit of (x + 3)/(x − 1) as x → 1. That gives a top of 4 and a bottom of 0. What should Bo conclude?
- 4/0: no finite limit
- substitute: the limit is 0
- 0/0: rewrite it first
3.
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
- substitute: the limit is 0
- 0/0: cancel (x + 3) first
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?
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?
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?
Pia is told that the limit of 3x² + kx as x → 2 is 20. What is k?
Ren is told that the limit of 2x² + kx as x → 3 is 12. What is k?
Tala is told that the limit of 5x² + kx as x → -1 is 9. What is k?
Zee is told that the limit of 4x² + kx as x → 2 is 6. What is k?

Limit laws and direct substitution. Every version has 10 questions, and the answer key prints on the second page.

Standard: none (beyond CCSS)

limit laws: Rules that let a limit pass through +, −, ×, ÷, powers and roots.

direct substitution: Putting x = a in, which works when the function is smooth at a.

0/0: Top and bottom both 0: rewrite before deciding.

k/0: A nonzero top over 0: the values grow without bound.

Combined the limits but dropped the numbers in front. The constant-multiple law keeps them: lim 3f = 3 lim f.

Subtracted a negative as if it were positive. Taking away −6 adds 6.

Squared a negative number and kept the minus. (−2)² = 4: a square is never negative.

What is the most common mistake with limit laws and direct substitution?

Combined the limits but dropped the numbers in front. The constant-multiple law keeps them: lim 3f = 3 lim f.

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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