Year: 2021
Topic: Analysis Calculus

Problem:
For every positive real number $x$, let \[ g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}. \] Find $\lim_{x \to \infty} \frac{g(x)}{x}$.

Solution:
By l'H\^opital's Rule, we have \begin{align*} &\lim_{r\to 0} \frac{\log((x+1)^{r+1}-x^{r+1})}{r} \\ &\quad = \lim_{r\to 0} \frac{d}{dr} \log((x+1)^{r+1}-x^{r+1}) \\ &\quad = \lim_{r\to 0} \frac{(x+1)^{r+1}\log(x+1)-x^{r+1}\log x}{(x+1)^{r+1}-x^{r+1}} \\ &\quad = (x+1)\log(x+1)-x\log x, \end{align*} where $\log$ denotes natural logarithm. It follows that $g(x) = e^{(x+1)\log(x+1)-x\log x} = \frac{(x+1)^{x+1}}{x^x}$. Thus \[ \lim_{x\to\infty} \frac{g(x)}{x} = \left(\lim_{x\to\infty}\frac{x+1}{x}\right) \cdot \left(\lim_{x\to\infty} \left(1+\frac{1}{x}\right)^x\right) = 1\cdot e = e. \] The limit is $\boxed{e}$.