Question

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

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

Answer 1

Answer: (D)

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