Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let L = limx→0
a−
√
a2−x2−x2
4
x4
, a > 0. If L is finite, then:</p>
a = 2
a = 1
L = 1/64
L = 1/32
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: Expand</p> \sqrt a2 -x2 using Binomial theorem: p a2 -x2 = a 1 -x2 a2 1/2 = a 1 -1 2 x2 a2 -1 8 x4 a4 -. . . = a -x2 2a -x4 8a3 -. . .<p><strong>2</strong>: Substitute into L:</p> L = lim x\to 0 a - a -x2 2a -x4 8a3 -x2 4 x4 = lim x\to 0 1 2a -1 4 x2 + x4 8a3 x4<p><strong>3</strong>: For L to be finite, the coefficient of x2 must be zero:</p> 1 2a -1 4 = 0 =\Rightarrow a = 2<p><strong>4</strong>: Then L =</p> 1 8a3 = 1 8 \times 8 = 1/64.
Correct Answer: (A, C)