Evaluating the limit $\lim \limits_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}$

Question

How do you evaluate the limit

$$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$$

You could try to evaluate the limit of the logarithm of this. — Did, Apr 25 '11 at 07:46

score 13 · Answer 1 · answered Apr 26 '11 at 04:14

13

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

answered Apr 26 '11 at 04:14

Alex Becker

60,569

+1 Very nice! I don't think you can get a simpler solution than this one. – Adrián Barquero Apr 26 '11 at 04:24

score 7 · Accepted Answer · answered Apr 25 '11 at 07:53

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?

Evaluating the limit $\lim \limits_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}$

2 Answers2