Question

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

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

Answer 1

Answer: (D)

$\text{ Let }y=\underset{y\to 0}{\lim }{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/x}$
$\Rightarrow \log y=\underset{x\to 0}{\lim }\frac{2}{x}\log \left(\frac{a^x+b^x+c^x}{3}\right)$
$\Rightarrow \log y=\underset{x\to 0}{\lim }\frac{2\log \left(a^x+b^x+c^x\right)-\log 3}{x}$
$\Rightarrow \log y=\log (abc{)}^{2/3}$
$\therefore [\text{ using L’Hospital’s rule }]$
$\therefore y=(abc{)}^{2/3}$