Let for , and let g(x)= cases x (1+ 1 x ),&0<x 1, 0,&x=0. cases Then _ n f ( 1n )f (…

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

f(x)=\left(1+\frac{1}{x}\right)^x

for

x>0

, and let

g(x)=\begin{cases}x\ln\left(1+\frac{1}{x}\right),&0<x\leq1,\\0,&x=0.\end{cases}

Then

\lim_{n\to\infty}\left\{f\left(\frac1n\right)f\left(\frac2n\right)f\left(\frac3n\right)\cdots f\left(\frac nn\right)\right\}^{1/n}

equals (1)

\sqrt{2}e

(2)

\sqrt{2e}

(3)

2\sqrt{e}

(4)

\sqrt{e}