Question

$\underset{x\to 0}{lim}{\left(\frac{a^x+b^x+c^x}{3}\right)}^{1/x}=$

Answer 1

$\underset{x\to 0}{lim}{\left(\frac{a^x+b^x+c^x}{3}\right)}^{1/x}$

We know that

${lim}_{x\to 0}f(x{)}^{g\left(x\right)}$ $=e^{{lim}_{x\to 0}\left[f\right(x)-1]g\left(x\right)}$

Therefore,

$\Rightarrow e^{\underset{x\to 0}{lim}(\frac{a^x+b^x+c^x}{3}-1)\times \frac{1}{x}}$

$\Rightarrow e^{\underset{x\to 0}{lim}\left(\frac{a^x+b^x+c^x-3}{3}\right)\times \frac{1}{x}}$

$\Rightarrow e^{\frac{1}{3}\times \underset{x\to 0}{lim}\left(\frac{a^x+b^x+c^x-3}{x}\right)}$

$\Rightarrow e^{\frac{1}{3}\times {lim}_{x\to 0}(\frac{a^x-1}{x}+\frac{b^x-1}{x}+\frac{c^x-1}{x})}$

$\Rightarrow e^{\frac{1}{3}\times (\ln a+\ln b+\ln c)}$

$\Rightarrow e^{\frac{1}{3}\times \left(\ln abc\right)}$

$\Rightarrow e^{\left(\ln {abc}^{\frac{1}{3}}\right)}$

$\Rightarrow {\left(abc\right)}^{\frac{1}{3}}$

Hence, this is the answer.