Limits of a function — A/L Combined Mathematics

How do I practise Limits of a function?

2 questions with worked solutions

1. What is the value of the limit lim_{x → 0} [sin(3x) / x]?
   1. A) 0
   2. B) 3
   3. C) 1
   4. D) 1/3
   Check my answer

   Correct answer: B) 3

   Rewriting the expression: lim_{x → 0} [sin(3x) / x] = 3 × lim_{x → 0} [sin(3x) / (3x)]. Since lim_{u → 0} [sin(u) / u] = 1, this equals 3 × 1 = 3.

   Show worked solution

   Step 1: Multiply and divide by 3: 3 · [sin(3x) / (3x)]. Step 2: Standard limit: lim_{u → 0} [sin u / u] = 1. Step 3: Value = 3 × 1 = 3. Answer: 3.
   Modelled on: the A/L Combined Mathematics examination pattern
2. Evaluate the limit lim_{x → 2} [(x² - 4) / (x - 2)].
   1. A) 2
   2. B) 8
   3. C) 4
   4. D) 0
   Check my answer

   Correct answer: C) 4

   Substituting x = 2 directly gives 0/0, so the expression must be simplified first. Factorising, x² - 4 = (x - 2)(x + 2), so for x ≠ 2 the expression equals x + 2. As x → 2 this tends to 2 + 2 = 4.

   Show worked solution

   Step 1: Direct substitution of x = 2 gives the indeterminate form 0/0, so factorise. Step 2: x² - 4 = (x - 2)(x + 2). Step 3: For x ≠ 2, (x² - 4) / (x - 2) = x + 2. Step 4: Limit = 2 + 2 = 4. Answer: 4.
   Modelled on: the A/L Combined Mathematics examination pattern

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Next in the syllabus

Where does Limits of a function fit in the syllabus?

Year 1 · Term 2