Why is $\,0\cdot\infty\,$ undefined?

Question

So, I was going through some calculus videos, and what I found was that it said that $\,0\cdot\infty\,$ is undefined. I'm confused why. Like, from an intuition stand point, it is obsviously $0$. Multiplication is simply repeated addition, and let's look at both sides. Let's say we are adding $\infty$ $0$ times to itself, the result is obviously $0$, as there is nothing to add, and it's simply blank, which means it's $0$. Let's take it the other way around, adding $0$ to itself an infinite number of times. This will again be $0$, as no matter how many you add $0$s together, it'll always remain $0$. So it clearly seems to be $0$, but why is it left undefined then?