# $\lim_{x \rightarrow 0} (\frac{a^x + b^x + c^x}{3})^{2/x}$ , a, b, c > 0 is equal to

Q: $\large \lim_{x \rightarrow 0} (\frac{a^x + b^x + c^x}{3})^{2/x}$ , a, b, c > 0 is equal to

(A) 0

(B) (abc)2/3

(C) abc

(D) (abc)1/2

Sol: Let $\large I = \lim_{x \rightarrow 0} (\frac{a^x + b^x + c^x}{3})^{2/x}$ ; (1 form )

$\large logI = \lim_{x \rightarrow 0} \frac{2}{x}log(\frac{a^x + b^x + c^x}{3})$

[Using L’Hospital]

$\large logI = \frac{2}{3} log(abc)$

$\large logI = log(abc)^{2/3}$

I = (abc)2/3

Hence (B) is the correct answer.