Word Count : 690

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आइए प्रत्येक कथन का विश्लेषण करके यह निर्धारित करें कि वे सत्य हैं या असत्य। --- ### (i) \(\lim_{x\rightarrow0}\frac{x^{2}\sin\frac{1}{x}}{\sin x}=1\) #### *हल:* इस सीमा को हल करने के लिए, हम संख्यात्मक और सैद्धांतिक दोनों तरीकों से मूल्यांकन करेंगे। - चूँकि \(\sin \frac{1}{x}\) का मान \([-1,1]\) के बीच दोलित करता है, इसलिए \[ -x^2 \leq x^2 \sin \frac{1}{x} \leq x^2 \] और \(\sin x \approx x\) के लिए छोटे \(x\) के आस-पास, यह समीकरण \[ \frac{x^2 \sin \frac{1}{x}}{\sin x} = \frac{x^2 \sin \frac{1}{x}}{x} \] \[ = x \sin \frac{1}{x} \] बनता है, जिसका मान \([-x, x]\) के भीतर दोलित करता है। - जब \(x \to 0\), तो \(\sin \frac{1}{x}\) के कारण यह सीमा निश्चित नहीं होती, __________ ____ ________ _______ ________ _________ _____.
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