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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?

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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"

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The aymptotes show the tendency of the pole movements in open loop.

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