Word Count : 575

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इस फलन \( f(x, y) \) को दिया गया है: \[ f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases} \] यहां पर \( f(x, y) \) के \( (0, 0) \) पर सांतत्य और अवकलनीयता की जाँच करनी है। ### 1. सांतत्य (Continuity) की जाँच: हम \( f(x, y) \) की सांतत्य जाँच करने के लिए यह जांचते हैं कि क्या: \[ \lim_{(x,y) \to (0,0)} f(x,y) = f(0,0) \] जहां \( f(0,0) = 0 \)। अब हमें यह देखना है कि क्या \( (x,y) \to (0,0) \) करते समय \( f(x,y) \) का मान \( 0 \) के बराबर आ रहा है। हम पॅथ-निर्भर सीमा की जाँच करेंगे। हम \( x = 0 \) और \( y = 0 \) के लिए विशेष पथों पर सीमा __________ ___ ____ ________ ________ ______.
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