Is infinity times zero = zero?

[tosh]

https://brilliant.org/wiki/is-infinity-times-zero-zero/

[Erlangen]

I haven't touched analytical mathematics for a very long time, so my understanding is quite rough now. Though the solution seems obvious, I hope the proof gives more explanation.

1. Is it OK to replace 0 with a function that approaches 0 at infinity? What rules are in place for substitution?

2. Is lim f(x) * lim g(x) equal to lim f(x)*g(x) ?

[simonblack → Erlangen]

1. No. In my opinion, zero can never equal non-zero.

Otherwise you could be 'only a little bit' pregnant.

[yongjik]

This website presents a question that can be only answered by precisely defining "infinity", and then shows an "answer" that just assumes (without proof) a horde of stuff, including "lim (f * g) = (lim f) * (lim g)".

This is not a good way to learn math.

[simonblack]

    infinity = 1+1+1+1 ..... +1+1

    infinity x 0 = (1x0)+(1x0)+(1x0) .... +(1x0)+1x0)

                 =  0   + 0   + 0    .... +  0  +  0

                 =  0

[simonblack]

Proof: We know ....

That is a non-proof. Test it by finishing the sentence differently, as in "Proof: We know ... that the world is flat."