Let $$\operatorname{rad}(x)=\prod_{\substack{p\mid x\\p\text{ prime}}}p,$$ denote the product of all prime divisors of a positive integer $x$. For example, we have $rad(10)=\prod_{p|10}p=2×5=10$ and $rad(q)=\prod_{p|q}p=q$ for all prime numbers $q$.
My question is about the behavior of $rad(x)=\prod_{p|x}p$ when $x$ goes to infinity.
There is no concisely describable "behavior." As $x$ increases, it will serially encounter integers. Some of those will be square-free, for which $rad(x)=x$, and some of those will be perfect powers of a single prime, for which $rad(x)=rad(p^k)=p$. It will most frequently encounter numbers that are neither square-free nor powers of a single prime, in which case $rad(x)<x$, but will be dependent on the idiosyncratic factorization of $x$, which is not systematic with increasing $x$.
In the first instance, as $x$ goes to infinity, its radical (being $x$ itself) goes to infinity as well. In the second instance, as $x=p^k$ goes to infinity, which is to say as $k$ goes to infinity, its radical remains unchanged at $p$. In the third case, it is hard to say anything at all about the magnitudes of the values that $rad(x)$ might take, save that those values must by definition be square-free numbers, and that in no case does $rad(x)$ exceed $x$.