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