Question:
$\lim_{x\rightarrow0}\frac{\sin(\pi \sin^{2}x)}{x^{2}} = $ ________.
$\lim_{x\rightarrow0}\frac{\sin(\pi \sin^{2}x)}{x^{2}} = $ ________.
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Multiply and divide by the inner function to match the standard limit form.
Updated On: Jun 26, 2026
- $\frac{\pi}{2}$
- $\pi$
- $2\pi$
- $\pi^{2}$
- $\frac{\pi^{2}}{2}$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Use the standard limit $\lim_{\theta\rightarrow0} \frac{\sin\theta}{\theta} = 1$.
Step 2: Meaning
Rewrite as $\lim_{x\rightarrow0} \frac{\sin(\pi \sin^2 x)}{\pi \sin^2 x} \times \frac{\pi \sin^2 x}{x^2}$.
Step 3: Analysis
The first part goes to 1 as $x\rightarrow0$. The second part is $\pi \times (\lim_{x\rightarrow0} \frac{\sin x}{x})^2 = \pi \times 1^2$.
Step 4: Conclusion
The final limit is $\pi$. Final Answer: (B)
Use the standard limit $\lim_{\theta\rightarrow0} \frac{\sin\theta}{\theta} = 1$.
Step 2: Meaning
Rewrite as $\lim_{x\rightarrow0} \frac{\sin(\pi \sin^2 x)}{\pi \sin^2 x} \times \frac{\pi \sin^2 x}{x^2}$.
Step 3: Analysis
The first part goes to 1 as $x\rightarrow0$. The second part is $\pi \times (\lim_{x\rightarrow0} \frac{\sin x}{x})^2 = \pi \times 1^2$.
Step 4: Conclusion
The final limit is $\pi$. Final Answer: (B)
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