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Grade 12 · Math

Evaluate a rational limit with a removable hole

Use factorization to evaluate a limit even when direct substitution gives 0/0.

6 authored questions + worked example

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Worked example

Find the limit as x approaches 2 of (x2 − 4)/(x − 2).

Answer & why

Answer: 4

Factor x2 − 4 as (x − 2)(x + 2). For x ≠ 2, the quotient is x + 2, whose limit is 2 + 2 = 4. The original quotient is undefined at 2.

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All questions

All 6 authored questions for this skill.

Question 1

Find the limit as x approaches 4 of (x2 − 16)/(x − 4).

  1. 10
  2. 8
  3. 7
  4. 9
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 8

Factor x2 − 16 as (x − 4)(x + 4). For x ≠ 4, the quotient is x + 4, whose limit is 4 + 4 = 8. The original quotient is undefined at 4.

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Question 2

Find the limit as x approaches 5 of (x2 − 25)/(x − 5).

  1. 12
  2. 9
  3. 11
  4. 10
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 10

Factor x2 − 25 as (x − 5)(x + 5). For x ≠ 5, the quotient is x + 5, whose limit is 5 + 5 = 10. The original quotient is undefined at 5.

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Question 3

Find the limit as x approaches 6 of (x2 − 36)/(x − 6).

  1. 13
  2. 11
  3. 14
  4. 12
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 12

Factor x2 − 36 as (x − 6)(x + 6). For x ≠ 6, the quotient is x + 6, whose limit is 6 + 6 = 12. The original quotient is undefined at 6.

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Question 4

Find the limit as x approaches 7 of (x2 − 49)/(x − 7).

  1. 15
  2. 16
  3. 14
  4. 13
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 14

Factor x2 − 49 as (x − 7)(x + 7). For x ≠ 7, the quotient is x + 7, whose limit is 7 + 7 = 14. The original quotient is undefined at 7.

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Question 5

Find the limit as x approaches 2 of (x2 − 4)/(x − 2).

  1. 5
  2. 3
  3. 6
  4. 4
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 4

Factor x2 − 4 as (x − 2)(x + 2). For x ≠ 2, the quotient is x + 2, whose limit is 2 + 2 = 4. The original quotient is undefined at 2.

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Question 6

Find the limit as x approaches 3 of (x2 − 9)/(x − 3).

  1. 5
  2. 7
  3. 8
  4. 6
A hint

Factor and cancel only for x away from the approach value; a limit concerns nearby values.

Answer & why

Answer: 6

Factor x2 − 9 as (x − 3)(x + 3). For x ≠ 3, the quotient is x + 3, whose limit is 3 + 3 = 6. The original quotient is undefined at 3.

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