Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>If limx→0
x(1+a cos x)−b sin x
x3
= 1, then:</p>
a = -5/2, b = -3/2
a = 5/2, b = 3/2
a = 5, b = 2
a = 2, b = 5
Step-by-Step Solution
Key Concept: General
<p>Expand cos x \approx1 -x2/2 and sin x \approxx -x3/6. Numerator: x(1 + a(1 -x2/2)) -b(x -x3/6) =</p> (1 + a -b)x + (b/6 -a/2)x3. For limit to exist, 1 + a -b = 0 =\Rightarrow b -a = 1. Limit is b/6 -a/2 = 1 =\Rightarrow b -3a = 6. Solving: a = -5/2, b = -3/2.
Correct Answer: 4