$\lim\limits_{N \to +\infty} \sqrt{N+1} \log\left (1+\frac{x}{N+1}\right)$ I have to compute the limit
$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as
$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively, this limit is 0, but I have no clue on how to solve it. Can you help me?

If we are allowed to use L' Hospital:
$$\lim_{n\to \infty}\frac{\ln(1+\frac{x}{n+1})}{\frac{1}{\sqrt{n+1}}} = \lim_{n\to \infty}\frac{2(n+1)^{5/2}}{(n+1+x)(n+1)^2} \to 0$$
