You treated infinity like a single, defined value. It is not. Because of this, you can't just say that ∞/∞=1, because it pretty much never is (and when it does, you've usually just gone on a long excisable detour). In fact, infinity is best treated as a direction. All the infinities are in that direction, but some of them are farther along than others, and there's no way to tell just by looking at them whether one infinity is farther than another.
So how do we compare infinities, if we can't tell by looking at them? We look at what generated them. That is the entire purpose of L'Hôpital's rule.
But since your expression doesn't contain any variables, we can't use L'hôpital's rule, either. So where does that leave us? With one inexorable truth: 1/π will always be 1/π, as long as you do the same thing to both parts of the fraction. Multiplication by zero, by the way, would give us 0/0, which leads us right back to L'hôpital's rule.