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Directional Derivatives and Limits 
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#1
Mar3112, 10:16 AM

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). 


#2
Mar3112, 12:17 PM

P: 746

you can choose different paths, and if the paths aren't equal, the limit does not exist.



#3
Mar3112, 01:31 PM

P: 9




#4
Mar3112, 01:53 PM

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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. 


#5
Mar3112, 02:39 PM

P: 9

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. 


#6
Mar3112, 04:39 PM

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P: 18,036

Try to approach is coming from a direction of the xaxis and of the yaxis. Are these two limits the same?



#7
Mar3112, 05:08 PM

P: 9




#8
Mar3112, 05:57 PM

P: 9

Could I just simply show that the directional derivative at (a1,b) in the direction of (a+1,b) is different from the directional derivative at (a1,b+1) in the direction of (a+1,b+1)?



#9
Mar3112, 08:20 PM

P: 8

That does not resolve my issue.



#10
Mar3112, 08:32 PM

P: 9




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