whats the proof for $\lim_{x → 0} [(a_1^x + a_2^x + .....+ a_n^x)/n]^{1/x} = (a_1.a_2....a_n)^{1/n}$ This equation is directly given in my book and I am don't know anything about its proof.I tried L'Hospital rule by differentiating the both numerator as well as denominator(division rule), but the result is still coming in indeterminate forms.I am a beginner , and haven't practiced limits that much. This formula is really confusing me. 
 A: You can compute the limit of the logarithm of your expression:
$$
\lim_{x\to0}\frac{\log(a_1^x+\dots+a_n^x)-\log n}{x}
$$
which is the derivative at $0$ of
$$
f(x)=\log(a_1^x+\dots+a_n^x)
$$
Since
$$
f'(x)=\frac{a_1^x\log a_1+\dots+a_n^x\log a_n}{a_1^x+\dots+a_n^x}
$$
we have
$$
f'(0)=\frac{\log a_1+\dots+\log a_n}{n}=
\log\bigl((a_1\dotsm a_n)^{1/n}\bigr)
$$
and therefore your limit is
$$
\lim_{x\to0}\left(\frac{a_1^x + a_2^x +\dots+ a_n^x}{n}\right)^{\!1/x} =e^{f'(0)}=(a_1\dotsm a_n)^{1/n}
$$
A: I suppose that all the $a_k$ are positive. Let $u_n(x)$ be your expression. Put $\displaystyle v_n(x)=\log u_n(x)=\frac{\log(\frac{a_1^x+\cdots+a_n^x}{n})}{x}$ and 
$\displaystyle F(x)= \log(\frac{a_1^x+\cdots+a_n^x}{n})=\log G(x)$. Then we have $\displaystyle v_n(x)=\frac{F(x)-F(0)}{x}$. Now $G$ has a derivative at $x=0$, and $F$ also. So $v_n(x)$ has for limit the derivative of $F$ at $0$. As $\displaystyle F^{\prime}(x)=\frac{G^{\prime}(x)}{G(x)}$ and $G(0)=1$, this is $G^{\prime}(0)$. Now $a_k^x=\exp(x\log a_k)$, and it is easy to finish. 
A: The simplest route is to take logarithms. Let $$f(x) = \left(\frac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}{n}\right)^{1/x}$$ and we need to calculate the limit of $f(x)$ as $x \to 0$. Let $L$ be this desired limit then we have
\begin{align}
\log L &= \log\left(\lim_{x \to 0}f(x)\right)\notag\\
&= \lim_{x \to 0}\log f(x)\text{ (via continuity of log)}\notag\\
&= \lim_{x \to 0}\frac{1}{x}\log\frac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}{n}\notag\\
&= \lim_{x \to 0}\frac{1}{x}\cdot\left(\dfrac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}{n} - 1\right)\cdot\dfrac{\log\left(1 + \dfrac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}{n} - 1\right)}{\dfrac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}{n} - 1}\notag\\
&= \frac{1}{n}\lim_{x \to 0}\frac{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x} - n}{x}\notag\\
&= \frac{1}{n}\lim_{x \to 0}\sum_{k = 1}^{n}\frac{a_{k}^{x} - 1}{x}\notag\\
&= \frac{1}{n}\sum_{k = 1}^{n}\lim_{x \to 0}\frac{a_{k}^{x} - 1}{x}\notag\\
&= \frac{1}{n}\sum_{k = 1}^{n}\log a_{k}\notag\\
&= \log(a_{1}a_{2}\cdots a_{n})^{1/n}\notag
\end{align}
It now follows that $$L = (a_{1}a_{2}\cdots a_{n})^{1/n}$$ We have used the following two standard limits in the above derivation $$\lim_{x \to 0}\frac{\log(1 + x)}{x} = 1,\,\lim_{x \to 0}\frac{a^{x} - 1}{x} = \log a$$
A: $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\begin{align}
&\color{#f00}{%
\lim_{x \to 0}\bracks{%
\pars{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x} \over n}^{1/x}}} =
\exp\pars{\lim_{x \to 0}\bracks{{1 \over x}\,
\ln\pars{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x} \over n}}}
\\[3mm] = &\
\exp\pars{\lim_{x \to 0}{\bracks{%
a_{1}^{x}\ln\pars{a_{1}} + a_{2}^{x}\ln\pars{a_{2}} + \cdots + a_{n}^{x}\ln\pars{a_{n}}}/n \over
\bracks{a_{1}^{x} + a_{2}^{x} + \cdots + a_{n}^{x}}/n}}\qquad
\pars{~\mbox{L'H}\mathrm{\hat{o}\mbox{pital Rule}}~}
\\[3mm] = &\
\exp\pars{\ln\pars{a_{1}} + \ln\pars{a_{2}} + \cdots +\ln\pars{a_{n}} \over n} =
\color{#f00}{\pars{a_{1}a_{2}\ldots a_{n}}^{1/n}}
\end{align}
