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  4. The value of the displaystyle lim x arrow 0((ax+bx+cx/3))2 / x, (a, b, c>0) is

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

ManipalManipal 2014

Solution:

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