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निम्न दो (02) प्रश्नों के लिए निम्नलिखित पर विचार कीजिए:
\(\text{Let } f(x)= \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \ge 1 \end{cases} \\\)
\(\lim_{x \to 0} f'(x)\) किसके बराबर है?
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दिया गया है,
फलन इस प्रकार परिभाषित है:
\( f(x) = \begin{cases} x^3, & \text{if} \, |x| < 1 \\ x^2, & \text{if} \, |x| \geq 1 \end{cases} \)
हमें यह ज्ञात करना है:
\( \lim_{x \to 0} f'(x) \)
|x| < 1 के लिए, फलन f(x) = x3 है, इसलिए अवकलज है:
\( f'(x) = 3x^2 \)
अब, x के 0 तक पहुँचने पर अवकलज की सीमा की गणना करें:
\( \lim_{x \to 0} f'(x) = \lim_{x \to 0} 3x^2 = 0 \)
∴ \(\lim_{x \to 0} f'(x) \) का मान 0 है।
सही उत्तर विकल्प (c) है
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