# Directional Derivatives and Limits

by autarch
Tags: derivatives, directional, limits
 P: 9 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).
 P: 746 you can choose different paths, and if the paths aren't equal, the limit does not exist.
P: 9
 Quote by Woopydalan 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.

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P: 38,879

## Directional Derivatives and Limits

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.
P: 9
 Quote by HallsofIvy 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.
 Mentor P: 16,537 Try to approach is coming from a direction of the x-axis and of the y-axis. Are these two limits the same?
P: 9
 Quote by micromass 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).
 P: 9 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)?
 P: 8 That does not resolve my issue.
P: 9
 Quote by Cecilia48 That does not resolve my issue.
What issue is that?
 P: 9 bump

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