Question:
If \( f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}} \), then \( \lim_{x \to \infty} f(x) \) is equal to
If \( f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}} \), then \( \lim_{x \to \infty} f(x) \) is equal to
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For limits at infinity, dominant terms determine behavior.
Updated On: May 1, 2026
- \(1 \)
- \(2 \)
- \( \frac{1}{2} \)
- \(0 \)
- \( \infty \)
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The Correct Option is A
Solution and Explanation
Concept:
For large \( x \), bounded functions like \( \sin x, \cos x \) become negligible compared to \( x \).
Step 1: Approximate numerator.
\[ x - \sin x \approx x \]
Step 2: Approximate denominator.
\[ x + \cos^2 x \approx x \]
Step 3: Simplify.
\[ \sqrt{\frac{x}{x}} = 1 \]
Step 1: Approximate numerator.
\[ x - \sin x \approx x \]
Step 2: Approximate denominator.
\[ x + \cos^2 x \approx x \]
Step 3: Simplify.
\[ \sqrt{\frac{x}{x}} = 1 \]
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