Evaluating the limit $\lim \limits_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}$ How do you evaluate the limit 
$$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$$

How about using squeeze theorem? Try squeezing this as $0 \leq \frac{x^x}{(x+1)^{x+1}} \leq \frac{x^x}{x^{x+1}} = \frac{1}{x}$.
