What is the significance of asymptotes in the root locus of a linear time invariant system. Does it say that for that particular pole on which it is starting doesn't have a practical pole in the closed loop if the open loop gain is taken to infinity?
The poles of your root locus plot will move in the direction of those asymptotes when you increase your open-loop gain.
The asymptotes can (e.g.) help you understand if increasing (or decreasing) your gain will move your poles towards the stable (negative half for a continuous system) or unstable region in your root-locus plot.
P.S. Some software (like Matlab) will show you the gain needed for placing a particular location on the asymptote (which will also move the other poles!) see here under "Root Locus Design"