A) \[{{(abc)}^{3}}\]
B) \[abc\]
C) \[{{(abc)}^{1/3}}\]
D) None of these
Correct Answer: D
Solution :
Let \[y=\underset{x\to 0}{\mathop{\lim }}\,\,{{\left( \frac{{{a}^{x}}+{{b}^{x}}+{{c}^{x}}}{3} \right)}^{2/x}}\] \[=\underset{h\to 0}{\mathop{\lim }}\,h\,\sin 1/h=0\times (-1\le \sin 1/h\le 1)=0\] \[=2\,\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\log \,({{a}^{x}}+{{b}^{x}}+{{c}^{x}})-\log 3}{x}\] Now applying L-Hospital?s rule, we have \[\log y=\log \,{{(abc)}^{2/3}}\,\Rightarrow \,\,y={{(abc)}^{2/3}}\]
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