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 Dedekind zeta function of the number field $K$. And is proven in a similar way to the classical result. Here is a reference: "The probability of relative primality of Gaussian integers". 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 Catalan constant.

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 Dedekind zeta function of the number field $K$. And is proven in a similar way to the classical result. Here is a reference: "The probability of relative primality of Gaussian integers". 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 Catalan constant.