I understand that when you have a limit with a product going to 0*inf, you cannot determine the limit yet, but what about when the product goes to inf*inf? Are we allowed to say that that is infinity, or is it still indeterminate like 0*inf? Thank you!

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It's just infinity. Think of like lim as x goes to inf of x^2. That's just like the lim of x*x, which is inf*inf. Or you could just think huge number times huge number is also huge; likewise inf*inf is inf (an even larger infinity, even).

That was my initial idea as well, but can it be proven? My teacher said that we can not just assume that.

Well, you could do an epsilon-delta proof if you're in an analysis class, but if you're taking like calc 1, 2, or 3 then you can definitely assume inf*inf goes to inf. I mean, for a bit of an explanation, inf*(anything)=inf as long as the anything is not 0 (sure, we can do negatives and make it -inf, but same idea). Furthermore, inf is not 0, it's not even a number, so inf*inf must be inf.