For every positive real number $x$, let
$$g(x)=\lim\limits_{r\to0}((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.$$\mathrm{Find}\lim_{x\to\infty}\frac{g(x)}{x}. The original answer is in the form of $e$, please provide the integer part of the result.

Let's think step by step and output the final answer within \boxed{}.
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