Grade 12 · Math · Calculus

12th grade Limits that need factoring or rationalizing 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

Limits that need factoring or rationalizing

Grade 12 Math Worksheet

NameDate

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

Ari wants the limit of (√(x + 5) − 3)/(x − 4) as x → 4. Direct substitution fails. What is the limit?
Bo wants the limit of (√(x + 12) − 4)/(x − 4) as x → 4. Direct substitution fails. What is the limit?
Esme wants the limit of (√(x − 3) − 2)/(x − 7) as x → 7. Direct substitution fails. What is the limit?
Jun wants the limit of (√(x + 21) − 5)/(x − 4) as x → 4. Direct substitution fails. What is the limit?
Kira wants the limit of (x³ − 64)/(x − 4) as x → 4. Direct substitution fails. What is the limit?
Milo wants the limit of (x³ − 216)/(x − 6) as x → 6. Direct substitution fails. What is the limit?
Noah wants the limit of (x³ + 64)/(x + 4) as x → -4. Direct substitution fails. What is the limit?
Pia wants the limit of (1/x − 1/2)/(x − 2) as x → 2. Direct substitution fails. What is the limit?
Ren wants the limit of (1/x − 1/3)/(x − 3) as x → 3. Direct substitution fails. What is the limit?
Tala wants the limit of (1/x − 1/4)/(x − 4) as x → 4. Direct substitution fails. What is the limit?

Limits that need factoring or rationalizing. Every version has 10 questions, and the answer key prints on the second page.

I can factor and cancel to find a 0/0 limit.
I can use the conjugate to clear a square root, and combine fractions in a limit.
I can choose the first step for a limit and find the constant that makes it exist.

Standard: none (beyond CCSS)

factor and cancel: Split top and bottom into factors and remove the shared one.

conjugate: √a + b for √a − b; their product has no square root.

difference quotient: (s(t) − s(a))/(t − a): an average speed between a and t.

Read 0/0 as 0. 0/0 means the top and bottom share a factor; cancel it, then substitute again.

Read 0/0 as 1. 0/0 has no value of its own; cancel the shared factor first.

Dropped part of the new bottom after using the conjugate. √(x + 4) + 2 at x = 0 is 2 + 2.

What is the most common mistake with limits that need factoring or rationalizing?

Read 0/0 as 0. 0/0 means the top and bottom share a factor; cancel it, then substitute again.

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