Why is there no L'Hopital's rule for functions of more than one variable

[uteskier]

I asked this question to my professor but couldn't give me a definitive answer.

My textbook says that if you were to take the Lim (x,y) --> (a,b) of f(x,y) there would be an infinite amount of paths it could take to approach (a,b). It then gives an example of some function where they take the Lim (x,y) --> (0,0) of f(x,y) . First they let x = 0 and take the limit as y approaches 0 (tracing the y axis). Then they do the same thing for x and take the limit so that is gets to the origin by tracing the x axis.

They only showed two ways to get to the origin for this function of two variables and I cant think of any other way to get to the origin. If they held x at some constant other than zero and took the limit as y approached zero, it would never hit the origin. Y may get to zero but x is still constant, so how are there an infinite amount of ways to get there?

When your dealing with a function of one variable it is bounded by one path.

If you were to try and apply L'Hopital's rule to a rational function of more than one variable, and took the partial derivative of top and bottom you would get four combinations fx(x,y)/fx(x,y) , fy(x,y)/fx(x,y) , fy(x,y)/fy(x,y) , fx(x,y)/fy(x,y) . Assuming you only had to apply L'Hopital's rule once and the limit of all of these 4 combinations approached the same L, would that not work? It may be a lot of work to take the limit 4 times, especially if you had to apply it twice and do it more.

I realize its a lot to unpack but its got me curious as to why that wouldn't work. Let me know where/if my logic is flawed

 

[uteskier]

To add on, if we were able to convert the equation to cylindrical components so that it becomes a function of just r on top and bottom, could we then apply L'Hopital's Rule?

 

[Steven G]

1st of all what makes you think that the only path to the origin is along the x or y axis? You really need to think more clearly. ANY curve that goes though the origin will work in these limits. y=x^2 goes through the origin so why can't you go to the origin along that path? You can. How about y=x and y=3x and y=mx for ANY m different from 0.

Now do you know how you would calculate the limit of a function of x and y as (x,y)-->(0,0) along the curve y=2x?? And yes, y=2x is a curve. I will answer that one for you. Just compute the limit as x-->0 of f(x,2x). Can you handle that?

Now onto L'Hopital's Rule: Before I engage in any conversation with you about what you said in your post I want to be sure that you know the proof of this theorem. Then we can talk about why it should follow in the 3-D case. Please post back.

 

[Steven G]

To add on, if we were able to convert the equation to cylindrical components so that it becomes a function of just r on top and bottom, could we then apply L'Hopital's Rule?

What do you think about your question. At this forum we try very had to get you the student to think through your problems while we give some hints. Please think about your question and try to argue why it will or will not work. This is a good question.

 

[uteskier]

1st of all what makes you think that the only path to the origin is along the x or y axis? You really need to think more clearly. ANY curve that goes though the origin will work in these limits. y=x^2 goes through the origin so why can't you go to the origin along that path? You can. How about y=x and y=3x and y=mx for ANY m different from 0.

Now do you know how you would calculate the limit of a function of x and y as (x,y)-->(0,0) along the curve y=2x?? And yes, y=2x is a curve. I will answer that one for you. Just compute the limit as x-->0 of f(x,2x). Can you handle that?

Now onto L'Hopital's Rule: Before I engage in any conversation with you about what you said in your post I want to be sure that you know the proof of this theorem. Then we can talk about why it should follow in the 3-D case. Please post back.

I definitely will post back. I have three exams this week and this topic wont be covered on the calc exam I have. When I have time in a few days I will get back to you on this topic after spending some time thinking about what you said. Thanks!

 

[Steven G]

Good luck on your exams.
As a math major I had to study for my exams plus study math concepts that would not be on the exams.

 

[uteskier]

Ive had some time to look at it more and Im starting to understand why it wont work. When I was thinking of a function of one variable I was thinking of some line defined by an equation. I don't know where my head was at when I wrote this but a function of two variables would define a plane in 3-Space. It totally makes sense how you can approach a point on that plane an infinite number of ways.

Below I have attached an attempt at a proof (It may be completely wrong haha) but I am failing to see difference between the two for their respective functions. Im sure I am just not interpreting it correctly or made a mistake.