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 sequence 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 sequence
converging to zero keeps changing its sign without settling to 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 sequence also
converges to 0. If L is not zero (for instance if it is infinite),
then the original sequence does not have a limit.