Does it tell you from which side it's approaching 0? Don't think so.

lim x->∞ (2/x) > lim x->∞ (1/x) for x > 0

lim x->∞ (2/x) - lim x->∞ (1/x) > 0 for x > 0

lim x->∞ (2/x - 1/x) > 0 for x > 0

lim x->∞ (1/x) > 0 for x > 0

1/x > 0 for x > 0

That's a tautology.

Also if we do the same for -1/x it approaches 0 from the other side. So... you've gained nothing.

I think you are fighting to preserve your use of infinity-limits