Question

Assertion :If $a_1,a_2,a_3,a_4,..........a_n>0$, then $\underset{x\to \infty }{lim}{\left\{\frac{a_1^{1/x}+a_2^{1/x}+a_3^{1/x}+....+a_n^{1/x}}{n}\right\}}^{nx}={\Pi }_{i=a}^n{a}_i$ Reason: If $\underset{x\to a}{lim}f\left(x\right)\to 1,\underset{x\to a}{lim}\left(x\right)\to \infty$, then $\underset{x\to a}{lim}{\left\{f\right(x\left)\right\}}^{g\left(x\right)}=e^{\underset{x\to a}{lim}\left(f\right(x)-1).g\left(x\right)}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\underset{x\to \infty }{lim}{\left[\frac{a_1^{\frac{1}{x}}+a_2^{\frac{1}{x}}+a_3^{\frac{1}{x}}+...a_n^{\frac{1}{x}}}{n}\right]}^{nx}=\underset{y\to 0}{lim}{\left[\frac{a_1^y+a_2^y+a_3^y+...+a_n^y}{n}\right]}^{\frac{n}{y}}$
$=exp\left[\underset{y\to 0}{lim}\frac{n}{y}[\frac{a_1^y+a_2^y+...a_n^y}{n}-1]\right]=exp\left[\underset{y\to 0}{lim}\left(\frac{a_1^y+a_2^y+...a_n^y-n}{y}\right)\right]$
$=exp\left[\underset{y\to 0}{lim}(\frac{a_1^y-1}{y}+\frac{a_2^y-1}{y}+...+\frac{a_n^y-1}{y})\right]$
$=e^{(\log a_1+\log a_2+...+\log a_n)}=e^{\log (a_1.a_2...a_n)}=a_1.a_2...a_n$