The value of lim from limx→∞ a11/x+a21/x+…+an1/xnnx, is

A
$a_{1} + a_{2} + \dots + a_{n}$
B
$e^{a_{1}} + a_{2} +_+ a_{A}$
C
$\frac{a_{1} + a_{2} + \dots + a_{n}}{n}$
D
$a_{1} a_{2} \dots \dots a_{n}$

Solution:

Let $x = \frac{1}{y}$ . Then,

$lim_{x \to \infty} {\{\frac{a_{1}^{1 / x} + a_{2}^{1 / x} + \dots + a_{n}^{1 / x}}{n}\}}^{n x}$

$\begin{array}{l} = lim_{y \to 0} {\{\frac{a_{1}^{y} + a_{2}^{y} + \dots . + a_{n}^{y}}{n}\}}^{n / y} \\ = lim_{y \to 0} {\{1 + \frac{a_{1}^{y} + a_{2}^{y} + \dots . + a_{n}^{y} - n}{n}\}}^{n / y}\} \\ = lim_{y \to 0} {\{1 + \frac{(a_{1}^{y} - 1) + (a_{2}^{y} - 1) + \dots . + (a_{n 1}^{y} - 1)}{n}\}}^{n / y} \\ = e^{lim_{y \to 0}} \{\frac{a_{1}^{y} - 1}{y} + \frac{a_{2}^{y} - 1}{y} + \dots . + \frac{a_{n}^{y} - 1}{y}\} \\ = e^{\log a_{1} + \log a_{2} + i \dots + \log a_{n}} = e^{\log (a_{1} a_{2} - a_{n})} = a_{1} a_{2} a_{3} \dots a_{n} \end{array}$

The value of lim from $lim_{x \to \infty} {\{\frac{a_{1}^{1 / x} + a_{2}^{1 / x} + \dots + a_{n}^{1 / x}}{n}\}}^{n x},$ is

Solution:

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