mathop {lim }limits_{x to 0} {left( {frac{{{a | vedclass

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(D) ધારો કે $y = \mathop {\lim }\limits_{x 	o 0} {\left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} ight)^{2/x}}$.બંને બાજુ પ્રાકૃતિક લઘુગણક લેતા:$\ln y =

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$(abc)^{2/3}$

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(D) ધારો કે $y = \mathop {\lim }\limits_{x 	o 0} {\left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} ight)^{2/x}}$.બંને બાજુ પ્રાકૃતિક લઘુગણક લેતા:$\ln y = \mathop {\lim }\limits_{x 	o 0} \frac{2}{x} \ln \left( \frac{a^x + b^x + c^x}{3} ight)$.આ $\frac{0}{0}$ સ્વરૂપ હોવાથી,આપણે $L$'$H$ôpital ના નિયમનો ઉપયોગ કરીએ છીએ:$\ln y = 2 \mathop {\lim }\limits_{x 	o 0} \frac{\frac{d}{dx} \ln \left( \frac{a^x + b^x + c^x}{3} ight)}{\frac{d}{dx} (x)}$.$\ln y = 2 \mathop {\lim }\limits_{x 	o 0} \frac{3}{a^x + b^x + c^x} \cdot \frac{a^x \ln a + b^x \ln b + c^x \ln c}{3}$.જ્યારે $x 	o 0$,ત્યારે $a^x, b^x, c^x 	o 1$:$\ln y = 2 \cdot \frac{1}{3} (\ln a + \ln b + \ln c) = \frac{2}{3} \ln(abc) = \ln((abc)^{2/3})$.તેથી,$y = (abc)^{2/3}$.

$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} \right)^{2/x}}$ જ્યાં $(a, b, c > 0)$ નું મૂલ્ય શું છે?

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