According to calculus, (inifnity*0) is actually what is considered an indeterminate form. Which means that if cannot have an answer. If attempt to find an answer you can get very many different things.
lim n->∞ (n * sin(1/n))
= lim n->0 sin(n) / n
= 1
The explanation if this is that sin of 1 /n as n approaches infinity turns out to approach 0. so then n is infinity as n approaches infinity and if you were to evaluate that limit you get infinity*0 = 1. But, then you go and try
lim n->∞ ln(n/(n+1))
= lim n->∞ ln( 1 / (1 + 1/n))
[and since 1/n goes to 0,]
= ln(1 / (1+0))
= ln(1)
= 0
Which also results, while evaluating the limit, as infinity*0 = 1 in this case. The point is there isn't an answer
