Directional Derivatives and Limits

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Discussion Overview

The discussion revolves around the use of directional derivatives to demonstrate the non-existence of a limit for a two-variable function, specifically in the context of functions that are undefined at certain points. Participants explore the implications of discontinuities and the challenges posed by insufficient information about the functions involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to use the directional derivative of a function to show that the limit does not exist, given that the function is undefined at the point of interest.
  • Another participant suggests that choosing different paths to approach the limit can indicate non-existence if the limits along those paths differ.
  • A participant expresses concern that without more information about the functions involved, it is impossible to provide a definitive answer.
  • There is a correction regarding the notation used for the function, with a suggestion to clarify the definitions of the functions involved.
  • One participant acknowledges a mistake in their earlier notation and reiterates the need to show that the limit does not exist, despite the ambiguity of the functions.
  • Another participant proposes approaching the limit from the x-axis and y-axis to compare the results.
  • A participant inquires whether comparing directional derivatives from different points could demonstrate the limit's non-existence.

Areas of Agreement / Disagreement

Participants express differing views on the effectiveness of using directional derivatives and the necessity of additional information about the functions to resolve the question of limit existence. No consensus is reached on the best approach or the implications of the directional derivatives.

Contextual Notes

Limitations include the lack of specific information about the functions g and h near the point of interest, which affects the ability to draw conclusions about the limit's existence.

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How can I use the directional derivative of a two variable function to show that the limit does not exist? For example, suppose I have a function f(x,y)=g(x)/f(y) and g(a)=f(b)=0 and the limit as x and y go to a and b is 0. How would I use the directional derivative to show that the limit at (a,b) does not exist.

So far, I have tried to take the directional derivatives of the f(x,y) at points around the (a,b), but I feel this is inconclusive because nothing is known about the function itself, other than the fact that it is undefined at (a,b).
 
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you can choose different paths, and if the paths aren't equal, the limit does not exist.
 
Woopydalan said:
you can choose different paths, and if the paths aren't equal, the limit does not exist.

That does not resolve my issue. If the only information that I have is that g(x)/f(y) is discontinuous at (a,b), a directional derivative along different paths won't show anything due to the ambiguity of the function. Furthermore, if the limit does not exist at (a,b), then I cannot use the gradient at (a,b). At least, I don't think I can.
 
What you have written really does not make much since. You say "f(x,y)=g(x)/f(y)" so that you are using f both for a function of 1 variable and a function of two variables. I think what you intend would be better written "f(x,y)= g(x)/h(y)".

In any case, it is impossible to give an answer to your question without more information as to what f and g are like close to 0.
 
HallsofIvy said:
What you have written really does not make much since. You say "f(x,y)=g(x)/f(y)" so that you are using f both for a function of 1 variable and a function of two variables. I think what you intend would be better written "f(x,y)= g(x)/h(y)".

In any case, it is impossible to give an answer to your question without more information as to what f and g are like close to 0.

Good point and thank you. Also, I noticed that I have made a mistake. f(x,y) should equal
g(y)/(f(x) not g(x)/f(y). As far as providing more information, g(x)/h(y) represents any function that is undefined at a particular point. The only information that is given is that the line x=a is not in the domain, and the answer can be shown with a directional derivative, although any method is acceptable; however, I must show that the limit of f(x,y) at (a,b) does not exist.

As far as describing what f(x) and g(y) are like close to 0, I do not know, since f(x) and g(y) merely represent one variable functions, but nothing specific.
 
Try to approach is coming from a direction of the x-axis and of the y-axis. Are these two limits the same?
 
micromass said:
Try to approach is coming from a direction of the x-axis and of the y-axis. Are these two limits the same?

I don't understand what you mean, but the only thing I am trying to prove is that the limit as (x,y)->(a,b) does not exist for g(y)/h(x). I am trying to do this with directional derivatives, and the line x=a is not in the domain of f(x,y)=g(y)/h(x).
 
Could I just simply show that the directional derivative at (a-1,b) in the direction of (a+1,b) is different from the directional derivative at (a-1,b+1) in the direction of (a+1,b+1)?
 
http://www.infoocean.info/avatar2.jpg That does not resolve my issue.
 
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  • #10
Cecilia48 said:
http://www.infoocean.info/avatar2.jpg That does not resolve my issue.

What issue is that?
 
Last edited by a moderator:

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