Directional Derivatives and Limits

In summary, the conversation discusses using the directional derivative of a two variable function to show that the limit does not exist at a given point. The example given involves a function f(x,y) which is undefined at (a,b) and the limit as x and y approach a and b is 0. The speaker has tried taking directional derivatives at points around (a,b) but found it inconclusive. They also mention the ambiguity of the function and the fact that the gradient cannot be used in this case. Towards the end, they mention trying to prove the limit does not exist with directional derivatives and discuss different approaches, but the conversation ends without a clear resolution.
  • #1
autarch
9
0
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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  • #2
you can choose different paths, and if the paths aren't equal, the limit does not exist.
 
  • #3
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.
 
  • #4
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
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.
 
  • #6
Try to approach is coming from a direction of the x-axis and of the y-axis. Are these two limits the same?
 
  • #7
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).
 
  • #8
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)?
 
  • #9
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?
 
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1. What is a directional derivative?

A directional derivative is a measure of how a function changes in the direction of a given vector. It represents the instantaneous rate of change of the function at a specific point in that direction.

2. How is a directional derivative calculated?

To calculate a directional derivative, you first need to find the partial derivatives of the function with respect to each variable. Then, you can use the vector dot product to combine the partial derivatives with the direction vector. The resulting value is the directional derivative.

3. What is the difference between a directional derivative and a regular derivative?

A directional derivative measures the change of a function in a specific direction, while a regular derivative measures the change of a function in a specific point. In other words, a directional derivative takes into account both the slope and the direction, while a regular derivative only considers the slope.

4. What is the significance of directional derivatives in real-world applications?

Directional derivatives have many applications in fields such as physics, engineering, and economics. For example, they can be used to analyze the rate of change of temperature in a specific direction in a physical system, or to optimize the direction of a path for a vehicle traveling through varying terrain.

5. What is a limit in the context of directional derivatives?

In the context of directional derivatives, a limit refers to the value that the directional derivative approaches as the direction vector approaches a specific direction. It represents the maximum rate of change of the function in that direction.

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