No, because infinity times zero is still zero.
I doubt that. In fact: Infinity*Zero = undefined/not solvable but NOT Zero. Lim (X-> INF
) X * (1/X) = INF * ZERO = ONE but Lim (X-> INF
) (2*X) * (1/X) = INF * ZERO = TWO. Conclusion there is NO (exact) solution to Infinity*Zero; and most certainly it's not = ZERO.
Actually, also, in your notation, Lim (X-> INF
) X * exp
(-X) = INF * ZERO = ZERO.
Proof, by l'H˘pital's rule
: Lim (X-> INF
) X * exp(-X) = Lim (X-> INF
) X / exp(X) = Lim (X-> INF
) 1 / exp(X) = Lim (X-> INF
) exp(-X) = ZERO. QED.
INF * ZERO is one of the indeterminate forms
Just to round things out, I should also point out that one can have, as it were, INF * ZERO = INF.