Question

The value of $\displaystyle\lim_{x\to\infty}\left(\frac{a^{1/x}_{1}+a^{1/x}_{2} +...........+a^{1/x}_{n} }{n}\right)^{nx} a_{i }> 0, i = 1, 2, ...... n$, is

• A. $a_1 + a_2 + ........... + a_n$
• B. $e^{a_{1}+a_{2}+...a_{n}}$
• C. $\frac{a_{1}+a_{2}+... +a_{n}}{n}$
• D. $a_{1}a_{2}a_{3}.......a_{n}$

Answer 1

Answer: (D)

Putting $x = \frac{1}{y}$, we get
$L = limit = \displaystyle\lim_{y \to 0}\left(\frac{a^{y}_{1}+a^{y}_{2}+..... + a^{y}_{n}}{n}\right)^{n / y} \quad\left(\because x \to\infty y \to 0\right)$
$\therefore log_{e} L = \displaystyle\lim_{y \to 0} \frac{n}{y}. log_{e} \frac{1}{n} \left(a^{y}_{1}+a^{y}_{2}+..... + a^{y}_{n}\right) \left(\frac{0}{0}\right)$
$= n\displaystyle\lim_{y \to 0} \frac{\frac{\left(a^{y}_{1}\,log \,a_{1}+a^{y}_{2}\,log \,a_{2}+..... + a^{y}_{n}\,log \,a_{n}\right)}{a^{y}_{1}+a^{y}_{2}+..... + a^{y}_{n}}}{1}$
$= n. \frac{log\left(a_{1}a_{2}....a_{n}\right)}{n}$
$\therefore log \,L = log\left(a_{1}.a_{2}.....a_{n} \right)\Rightarrow L = a_{1}.a_{2}.a_{3}......a_{n}$