Evaluating $\lim_{n\rightarrow \infty} \frac{n\sqrt{\ln n}}{(n+1)\sqrt{\ln(n+1)}}$ I'm supposed to look at
$$\lim_{n\rightarrow \infty} \frac{n\sqrt{\ln n}}{(n+1)\sqrt{\ln(n+1)}}$$
which is of course the result of a ratio test. While Wolfram Alpha tells me the limit is $1$, I don't know how to simplify this expression any further to come to the same conclusion.
Any suggestions on how to proceed would be appreciated. 

We have
$$\dfrac{n}{n+1} \dfrac{\sqrt{\ln(n)}}{\sqrt{\ln(n+1)}} = \dfrac{n}{n+1} \dfrac{\sqrt{\ln(n)}}{\sqrt{\ln(n) + \ln(1+1/n)}} = \dfrac1{1+1/n} \dfrac1{\sqrt{1 + \dfrac{\ln(1+1/n)}{\ln(n)}}}$$
Now recall that
$$\lim_{n \to \infty} a_n b_n = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n$$
whenever $\lim_{n \to \infty} a_n$ and $\lim_{n \to \infty} b_n$ exist. Use this to compute what you want.
