Limits, Continuity & Differentiability
General
Grade None
Question:
<p>If limx→a 2x−
√
x2+3a2
√x+a−
√
2a =
√
2 (where a ∈R+), then a is equal to:</p>
1/3
1/2\sqrt{2}
1/3\sqrt{2}
1/9
Step-by-Step Solution
Key Concept: Identify the form of the limit (0/0) and apply L’Hopital’s Rule or rationalization to resolve the
indeterminacy.
<p><strong>1</strong>: Check form at x = a: Num = 2a -</p> \sqrt 4a2 = 0, Denom = \sqrt 2a - \sqrt 2a = 0. (Using a > 0).<p><strong>2</strong>: Apply L’Hopital’s Rule:</p> Num′ = 2 - 2x 2 \sqrt x2 + 3a2 = 2 - x \sqrt x2 + 3a2 x\to a ---\to 2 -a 2a = 3/2 Denom′ = 1 2\sqrt{x} + a x\to a ---\to 1 2 \sqrt 2a<p><strong>3</strong>: Limit =</p> 3/2 1/(2 \sqrt 2a) = 3 \sqrt 2a.<p><strong>4</strong>: Given 3</p> \sqrt 2a = \sqrt 2 =\Rightarrow 3\sqrt{a} = 1 =\Rightarrow \sqrt{a} = 1/3.<p><strong>5</strong>: Solving for a, we get a = 1/9.</p>
Correct Answer: 4