Question

The value of the $\underset{x\to 0}{\lim }{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/x}$, $(a,b,c>0)$ is

• A. $(abc{)}^3$
• B. $abc$
• C. $(abc{)}^{1/3}$
• D. None of these

Answer 1

Answer: (D)

Let $y=\underset{x\to 10}{\lim }{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/x}$
$\Rightarrow \log y=\underset{x\to 10}{\lim }\frac{2}{x}\log \left(\frac{a^x+b^x+c^x}{3}\right)$
$=\underset{x\to 10}{\lim }\frac{\log \left(a^x+b^x+c^x\right)-\log 3}{x}$
$\Rightarrow \log y=\log (abc{)}^{2/3}$ (by L-Hospital's rule)
$\Rightarrow y=(abc{)}^{2/3}$