<p>\(\lim_{x \to 2} \dfrac{\sqrt{2\sin^2(x-2)}}{x-2}\)</p>
Step-by-Step Solution
Key Concept: Recognize that √(sin²θ) = |sinθ|, and the limit depends on whether (x-2) approaches from left or right. Since |sin(x-2)|/(x-2) has different signs from each side, the two-sided limit does not exist.
<p><strong>Step 1:</strong> Simplify the expression using √(sin²(x-2)) = |sin(x-2)|</p><p>$$\lim_{x \to 2} \frac{\sqrt{2}|\sin(x-2)|}{x-2}$$</p><p><strong>Step 2:</strong> Analyze the right-hand limit (x → 2⁺, so x-2 > 0):</p><p>$$\lim_{x \to 2^+} \frac{\sqrt{2}\sin(x-2)}{x-2} = \sqrt{2} \cdot 1 = \sqrt{2}$$</p><p><strong>Step 3:</strong> Analyze the left-hand limit (x → 2⁻, so x-2 < 0):</p><p>$$\lim_{x \to 2^-} \frac{\sqrt{2}(-\sin(x-2))}{x-2} = \sqrt{2} \cdot \lim_{x \to 2^-} \frac{-\sin(x-2)}{x-2} = -\sqrt{2}$$</p><p><strong>Step 4:</strong> Since left and right limits are different (−√2 ≠ √2), the limit does not exist.</p><p>∴ Answer: B (Limit does not exist)</p>
Correct Answer: B