How do you evaluate the limit
$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$
How do you evaluate the limit
$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$
I think we should be witty about how we write it. How about we consider instead the limit $ \lim_{x \to \infty} \frac{x^x}{(x+1)^x (x+1)} = \lim_{x \to \infty} \left ( \frac{x}{x+1} \right )^{x} * \frac{1}{x+1} $
I think that this is suggestive of a proof?
How about using squeeze theorem? Try squeezing this as $0 \leq \frac{x^x}{(x+1)^{x+1}} \leq \frac{x^x}{x^{x+1}} = \frac{1}{x}$.