2021 Putnam A2

For every positive real number $x$, let

$$

g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}.

$$

Find $\lim_{x \to \infty} \frac{g(x)}{x}$.

对于每个正实数 $x$，令

$$

g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}。

$$

求 $\lim_{x \to \infty} \frac{g(x)}{x}$。