The limit of the ratio of two stochastic integrals

Question

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is a standard Brownian Motion. I just don't know what properties of Brownian motion to use to handle integral $\int_{0}^{\infty}B(s)dB(s)$ and $\int_{0}^{\infty}B^2(s)ds$.