Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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

1 Answer 1

Because the limit at 0 of a function defined on the real line exists if and only if its limit at 0 from the left and its limit at 0 from the right both exist and coincide. In particular, if its limit at 0 from the left and its limit at 0 from the right both exist but they differ, the limit itself does not exist.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.