A calculus problem by Hamza A

Calculus Level 3

$\large \lim_{x\to \infty} \left [ \dfrac{\left(1+ \frac1x \right)^x}e\right ]^x$

If the limit above can be expressed as $e^{-A/B}$ , where $A$ and $B$ are coprime positive integers, find $A+B$ .

1 solution

First Last
Jun 9, 2017

$L=\lim_{x\to\infty}x\ln{\frac{(1+\frac1{x})^x}{e}}$

$\quad\ln{L}=\lim_{x\to\infty}x^2\ln(1+\frac1{x})-x=$

$\lim_{x\to\infty}\frac{x\ln(1+\frac1{x})-1}{\frac1{x}}=\text{ apply L'Hopital's Rule twice and cleaning up:}$

$\ln{L}=\lim_{x\to\infty}\frac{-x^3}{2(x^3+2x^2+x)} = \frac{-1}{2}$

$L = \boxed{e^\frac{-1}{2}}$