MCQ
$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} \right)^{2/x}}$; $(a,\;b,\;c > 0) = . . .$
- A${(abc)^3}$
- B$abc$
- C${(abc)^{1/3}}$
- ✓એકપણ નહી.
$\Rightarrow$ $\log y = \mathop {\lim }\limits_{x \to 0} \, \frac{2}{x} \log \left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} \right)$
$ = 2\,\mathop {\lim }\limits_{x \to 0} \,\frac{{\log \,({a^x} + {b^x} + {c^x}) - \log 3}}{x}$
Now applying $L-$ Hospital’s rule, we have
$\log y = \log \,{(abc)^{2/3}}\, \Rightarrow \,\,y = {(abc)^{2/3}}$
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