suppose

Limf(x)=+∞ value

Limg(x)=+∞ value

Assuming that limf(x)g(x) is bounded, there must be a number n that makes LIMF (x) G (x).

And since the formula N*limf(x)=lin[N*f(x)]

Because limf(x)=+∞, Lin [n * f (x)] = +∞ ................ (1).

Because limg(x) is infinite and of course > n,

So LIMF (x) * LIMG (x) > LIMF (x) * n = lim [n * f (x)], compared with (1), it is greater than +∞, of course.