यदि f(0) = 2 है,तो mathop {lim }limits_{x to | vedclass

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(C) माना $L = \mathop {\lim }\limits_{x 	o 0} \frac{{\int\limits_0^x {tf(x)dt + \int\limits_0^x xf(t)dt} }}{{{x^2}}}$$L = \mathop {\lim }\limits_{x \

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$3$

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(C) माना $L = \mathop {\lim }\limits_{x 	o 0} \frac{{\int\limits_0^x {tf(x)dt + \int\limits_0^x xf(t)dt} }}{{{x^2}}}$$L = \mathop {\lim }\limits_{x 	o 0} \frac{{f(x) \int\limits_0^x t dt + x \int\limits_0^x f(t) dt}}{{{x^2}}}$$L = \mathop {\lim }\limits_{x 	o 0} \frac{{f(x) \cdot \frac{x^2}{2} + x \int\limits_0^x f(t) dt}}{{{x^2}}}$$L = \mathop {\lim }\limits_{x 	o 0} \left( \frac{f(x)}{2} + \frac{\int\limits_0^x f(t) dt}{x} ight)$दूसरे पद के लिए $L$'Hopital के नियम का उपयोग करने पर: $\mathop {\lim }\limits_{x 	o 0} \frac{\int\limits_0^x f(t) dt}{x} = \mathop {\lim }\limits_{x 	o 0} \frac{f(x)}{1} = f(0)$$L = \frac{f(0)}{2} + f(0) = \frac{3}{2} f(0)$चूंकि $f(0) = 2$ दिया गया है,$L = \frac{3}{2} 	imes 2 = 3$.

यदि $f(0) = 2$ है,तो $\mathop {\lim }\limits_{x \to 0} \frac{{\int\limits_0^x {\left( {tf(x) + xf(t)} \right)dt} }}{{{x^2}}}$ का मान ज्ञात कीजिए।

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