\(\displaystyle \lim_{x \to +\infty}\frac{x^x}{(x+1)^x}=\displaystyle \lim_{x \to +\infty}\left (\frac{x}{x+1} \right )^x=\displaystyle \lim_{x \to +\infty}\left (\frac{x+1}{x} \right )^{-x}\\=\displaystyle \lim_{x \to +\infty}\left (1+\frac{1}{x} \right )^{-x}=\frac{1}{\displaystyle \lim_{x \to +\infty}\left (1+\frac{1}{x} \right )^x }= \frac 1e \)