Question:
The value of $\lim_{x \to 0} \dfrac{\sin^2 x}{1 - \cos x}$ is equal to:
The value of $\lim_{x \to 0} \dfrac{\sin^2 x}{1 - \cos x}$ is equal to:
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Convert expressions using identities before applying limits.
Updated On: Apr 24, 2026
- $4$
- $2$
- $\frac{1}{2}$
- $\frac{1}{4}$
- $0$
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The Correct Option is B
Solution and Explanation
Concept:
• Use identity: \[ 1 - \cos x = 2\sin^2 \frac{x}{2} \]
Step 1: Rewrite expression
\[ \frac{\sin^2 x}{1 - \cos x} = \frac{\sin^2 x}{2\sin^2 \frac{x}{2}} \]
Step 2: Use identity
\[ \sin x = 2\sin \frac{x}{2} \cos \frac{x}{2} \] \[ \sin^2 x = 4\sin^2 \frac{x}{2} \cos^2 \frac{x}{2} \]
Step 3: Substitute
\[ \frac{4\sin^2 \frac{x}{2} \cos^2 \frac{x}{2}}{2\sin^2 \frac{x}{2}} = 2\cos^2 \frac{x}{2} \]
Step 4: Apply limit
\[ \lim_{x \to 0} 2\cos^2 \frac{x}{2} = 2 \] Final Conclusion:
\[ = 2 \]
• Use identity: \[ 1 - \cos x = 2\sin^2 \frac{x}{2} \]
Step 1: Rewrite expression
\[ \frac{\sin^2 x}{1 - \cos x} = \frac{\sin^2 x}{2\sin^2 \frac{x}{2}} \]
Step 2: Use identity
\[ \sin x = 2\sin \frac{x}{2} \cos \frac{x}{2} \] \[ \sin^2 x = 4\sin^2 \frac{x}{2} \cos^2 \frac{x}{2} \]
Step 3: Substitute
\[ \frac{4\sin^2 \frac{x}{2} \cos^2 \frac{x}{2}}{2\sin^2 \frac{x}{2}} = 2\cos^2 \frac{x}{2} \]
Step 4: Apply limit
\[ \lim_{x \to 0} 2\cos^2 \frac{x}{2} = 2 \] Final Conclusion:
\[ = 2 \]
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