The probability tends to $\frac{1}{\zeta(2)}=\frac{6}{\pi^2}$ as was mentioned by Qiaochu. This actually generalizes to arbitrary number fields, and is a less commonly known fact. 

In fact in any number field, the probability that two ideals are relatively prime is given by $1/\zeta_K(2)$, where $\zeta_K$ is the <a href="https://en.wikipedia.org/wiki/Dedekind_zeta_function">Dedekind zeta function</a> of the number field $K$. And is proven in a similar way to the classical result. Here is a reference: <a href="https://doi.org/10.1007/3-540-51084-2_23">"The probability of relative primality of Gaussian integers"</a>. For example the analogous probability for Gaussian integers is $6/(\pi^2G)$ where $G=1-\frac{1}{3^2}+\frac{1}{5^2}+\cdots$ is the <a href="https://en.wikipedia.org/wiki/Catalan_constant">Catalan constant</a>.