Directed Infinity for Optimization Problem limit

Hi all, I am solving a complex algebraic equation for the variable n. I solve for it, and get a analytical answer. I then take the limit of that answer as a variable tends to +infinity. When I solve this in mathematica, I get the following solution:

DirectedInfinity[ -B*ex-tx + sqrt( (B*ex+tx)^2 ) ] / L , where L is a constant.

Conventional mathematical wisdom tells me that the numerator cancels out and becomes 0. I'm confused. Any thoughts as to how this limit can be either +infinity or -infinity

For reference: DirectedInfinity[ any positive # ] = + inf

DirectedInfinity[ any negative # ] = - inf

DirectedInfinity[ 0 ] = UNKNOWN