# Directional Derivatives and Limits

Conceptually, why does a limit fail to exist at a point (a,b) if the directional derivative is not the same around the point (a,b)? Specifically, I mean the directional derivative is a different value as the point is approached from the left and is a different value as the point is approached from the right. Why does this mean that the limit does not exist?

P.S. I am looking for an extremely detailed explanation.

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You asked: "Why does this mean that the limit does not exist?" The limit of what? The directional derivatives themselves are limits and you just finished saying that they exist (with different values). Do you mean the limit of the original function at the point, rather than the limit of certain difference quotients at the point? – Dave L. Renfro Mar 30 '12 at 19:51