<p>If \(f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}}\) then \(\lim_{x \to \infty} f(x)\) is</p>
Step-by-Step Solution
Key Concept: Divide both numerator and denominator by x to extract dominant terms as x→∞; the constant terms become negligible compared to x, leaving only the x coefficients to determine the limit.
<p><strong>Step 1:</strong> Write the limit as:</p><p>$$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \sqrt{\frac{x - \sin x}{x + \cos^2 x}}$$</p><p><strong>Step 2:</strong> Divide numerator and denominator by x inside the square root:</p><p>$$= \lim_{x \to \infty} \sqrt{\frac{\frac{x - \sin x}{x}}{\frac{x + \cos^2 x}{x}}} = \lim_{x \to \infty} \sqrt{\frac{1 - \frac{\sin x}{x}}{1 + \frac{\cos^2 x}{x}}}$$</p><p><strong>Step 3:</strong> Apply limit laws. As $x \to \infty$:</p><p>• $\frac{\sin x}{x} \to 0$ (bounded function divided by unbounded)</p><p>• $\frac{\cos^2 x}{x} \to 0$ (bounded function divided by unbounded)</p><p><strong>Step 4:</strong> Evaluate:</p><p>$$= \sqrt{\frac{1 - 0}{1 + 0}} = \sqrt{1} = 1$$</p><p>∴ Answer: C (which equals 1)</p>
Correct Answer: C