Word Count : 515

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हम पहले प्रश्न खंड ख) को क्रमबद्ध रूप से हल करते हैं। --- (क) ध्रुवीय निर्देशांकों का प्रयोग करते हुए सीमा ज्ञात करना दी गई सीमा है: [ \lim_{\substack{x \to 0 \ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} ] ध्रुवीय निर्देशांक में, हम लेते हैं: [ x = r \cos\theta, \quad y = r \sin\theta, \quad r \to 0 ] तो भिन्नक (numerator) और हर (denominator) को लिखते हैं: [ x^3 - y^3 = (r \cos\theta)^3 - (r \sin\theta)^3 = r^3 (\cos^3\theta - \sin^3\theta) ] [ x^2 + y^2 = (r \cos\theta)^2 + (r \sin\theta)^2 = r^2 (\cos^2\theta + \sin^2\theta) = r^2 ] इसलिए सीमा बनती है: [ \frac{x^3 - y^3}{x^2 + y^2} = \frac{r^3 (\cos^3\theta - \sin^3\theta)}{r^2} = r (\cos^3\theta - \sin^3\theta) ] जब ( __________ _______ _____ ____ _______ ______ _______ __________ __________ ________.
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