What happens when we face the indeterminate product 0⋅∞, the "0" fails to be one-sided and "putting infinity under" failed to give an answer?

Since |0| = 0⁺, the indeterminate product |0|⋅∞ can be handled in the usual way and we can try to put the "|0|" part into the denominator. Of course, this can also fail, such problems then have to be handled individually.

So let's be optimistic and assume that we investigated the version |0|⋅∞ (that is, we put the part of the function that makes the zero into absolute value) and we found that it has a limit L (proper or improper). What conclusion can we make about the original problem?

The fact that the "0" part failed to be one-sided means that the part converging to zero keeps changing its sign without settling on one side. Therefore, when the version |0|⋅∞ is getting really close to L, the original expression 0⋅∞ is sometimes close to L, but sometimes also close to -L, and it keeps jumping from L to -L without settling on one of them. This should remind you of the oscillation-type problem. Such problem prevents limit from existing, unless of course the size of oscillation is zero, that is, L is zero.

Conclusion: If L = 0, then the original limit also converges to 0. If L is not zero (for instance if it is infinite), then the original limit does not exist.