Question:
\( \lim_{x \to 0} \frac{\sqrt{\cos 2x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \) is equal to
\( \lim_{x \to 0} \frac{\sqrt{\cos 2x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \) is equal to
Show Hint
For limits of type \( \frac{f(x)-f(0)}{x} \), directly use derivative at 0.
Updated On: Apr 21, 2026
- \( \frac{1}{4} \)
- \( -\frac{1}{4} \)
- \( \frac{1}{2} \)
- \( -\frac{1}{2} \)
- \( -1 \)
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The Correct Option is B
Solution and Explanation
Concept:
Use expansion and derivative form.
Step 1: At \( x=0 \), both terms equal.
Apply derivative form: \[ \lim_{x\to 0} \frac{f(x)-f(0)}{x} = f'(0) \]
Step 2: Differentiate expression.
After simplification: \[ = -\frac{1}{4} \]
Step 1: At \( x=0 \), both terms equal.
Apply derivative form: \[ \lim_{x\to 0} \frac{f(x)-f(0)}{x} = f'(0) \]
Step 2: Differentiate expression.
After simplification: \[ = -\frac{1}{4} \]
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