Grade 12 · Math · Calculus

12th grade Limits that need factoring or rationalizing worksheet

6 printable versionsBuilding Confidence10 questions · answer keyFree · no account

Level: Building Confidence · On Level · Challenge Me

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Limits that need factoring or rationalizing

Grade 12 Math Worksheet - Building Confidence

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Ari wants the limit of (x² + 3x − 10)/(x − 2) as x → 2. Direct substitution fails. What is the limit?
Bo wants the limit of (x² + 2x − 15)/(x − 3) as x → 3. Direct substitution fails. What is the limit?
Jun wants the limit of (x² − 7x + 10)/(x − 2) as x → 2. Direct substitution fails. What is the limit?
After multiplying by the conjugate, Kira's limit as x → 3 became 1/(√(x + 1) + 2). What is the limit?
After multiplying by the conjugate, Luz's limit as x → 0 became 1/(√(x + 9) + 3). What is the limit?
After multiplying by the conjugate, Milo's limit as x → 9 became 1/(√(x + 7) + 4). What is the limit?
After multiplying by the conjugate, Noah's limit as x → 9 became 1/(√(x + 16) + 5). What is the limit?
Ren has the factored form (x + 2)(x + 5)/((x + 2)(x + 4)). What is its limit as x → -2?
Tala has the factored form (x − 1)(x + 6)/((x − 1)(x − 3)). What is its limit as x → 1?
Zee has the factored form (x − 4)(x − 1)/((x − 4)(x + 2)). What is its limit as x → 4?

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.

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