Question

The value of the $\underset{x\to 10}{lim}{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/3}(a,b,c>0)$ is

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

Answer 1

The correct option is D None of these

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