Question

$\displaystyle \lim_{x \to 0}$ $\left(\frac{1^{x}+2^{x}+3^{x}+.....+n^{x}}{n}\right)^{\frac{a}{x}}$

• A. $\left(n!\right)^{\frac{a}{n}}$
• B. $\left(n!\right)^{\frac{n}{a}}$
• C. $\left(n!\right)^{an}$
• D. None of these

Answer 1

Answer: (A)

$\displaystyle \lim_{x \to 0}$ $\left(\frac{1^{x}+2^{x}+3^{x}+.....+n^{x}}{n}\right)^{\frac{a}{x}}\quad$ $(1^{\infty}$ form$)$
$=e^{\displaystyle \lim_{x \to 0}\left(\frac{1^{x}+2^{x}+...+n^{x}}{n}-1\right)\left(\frac{a}{x}\right)}$
$=e^{\displaystyle \lim_{x \to 0}\left[\left(\frac{1^{x}-1}{x}\right)+\left(\frac{2^{x}-1}{x}\right)+...+\left(\frac{n^{x}-1}{x}\right)\right] \frac{a}{n}}$
$=e^{log\left(1\cdot2 ... n\right)\cdot\frac{a}{n}}=e^{log\left(n!\right) \frac{a}{n}}$
$=\left(n!\right)^{\frac{a}{n}}$