Grade 12 · Math · Calculus

12th grade Limit laws and direct substitution worksheet

6 printable versionsOn Level10 questions · answer keyFree · no account

When it’s done, scan the sheet with your phone and PencilGo marks it and picks the next one. Create a free account

Version

Limit laws and direct substitution

Grade 12 Math Worksheet

NameDate

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

Ari knows lim f(x) = -2 as x → 1. What is the limit of [f(x)]^3 as x → 1?
Bo knows lim f(x) = 3 as x → 4. What is the limit of [f(x)]^2 as x → 4?
Esme knows lim f(x) = -2 as x → 0. What is the limit of [f(x)]^5 as x → 0?
Jun knows lim f(x) = -3 as x → 2. What is the limit of [f(x)]^3 as x → 2?
Kira finds the limit of (x² + 5)/(x + 4) as x → -2. The bottom is not 0 there. What is the limit?
Luz finds the limit of (x² − 1)/(x + 2) as x → 3. The bottom is not 0 there. What is the limit?
Milo finds the limit of (x² + 9)/(x + 3) as x → -1. The bottom is not 0 there. What is the limit?
Noah finds the limit of (x² + 1)/(x + 7) as x → -3. The bottom is not 0 there. What is the limit?
A bowl of soup cools in a 22 °C room: T(t) = 22 + 72(1/2)^(t/8) °C after t minutes. Ren asks where T heads as t → 16. What is that limit, in °C?
A mug of cocoa cools in an 18 °C room: T(t) = 18 + 64(1/2)^(t/5) °C after t minutes. Tala asks where T heads as t → 10. What is that limit, in °C?

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.

Every sheet you scan goes into your child’s record: dated, marked and ready for the year-end portfolio. Homeschool records