Question:
$\lim_{\theta\rightarrow 0}\frac{\theta \sin 2\theta}{1-\cos 2\theta} =$
$\lim_{\theta\rightarrow 0}\frac{\theta \sin 2\theta}{1-\cos 2\theta} =$
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Small angle approximations $(\sin \theta \approx \theta, \cos \theta \approx 1 - \frac{\theta^2}{2})$ can simplify limits involving trigonometric terms.
Updated On: Apr 28, 2026
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- $\frac{-1}{2}$
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- $\frac{1}{2}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Use the identity $1 - \cos 2\theta = 2\sin^2 \theta$ and the limit property $\lim_{x\rightarrow0} \frac{\sin x}{x} = 1$.
Step 2: Analysis
$\lim_{\theta\rightarrow0}\frac{\theta (2 \sin \theta \cos \theta)}{2 \sin^2 \theta} = \lim_{\theta\rightarrow0} \frac{\theta \cos \theta}{\sin \theta}$.
Step 3: Calculation
$\lim_{\theta\rightarrow0} (\frac{\theta}{\sin \theta}) \cdot \cos \theta = 1 \cdot 1 = 1$. Final Answer: (A)
Use the identity $1 - \cos 2\theta = 2\sin^2 \theta$ and the limit property $\lim_{x\rightarrow0} \frac{\sin x}{x} = 1$.
Step 2: Analysis
$\lim_{\theta\rightarrow0}\frac{\theta (2 \sin \theta \cos \theta)}{2 \sin^2 \theta} = \lim_{\theta\rightarrow0} \frac{\theta \cos \theta}{\sin \theta}$.
Step 3: Calculation
$\lim_{\theta\rightarrow0} (\frac{\theta}{\sin \theta}) \cdot \cos \theta = 1 \cdot 1 = 1$. Final Answer: (A)
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