Showing that a multivariable limit does not exist

In summary, the conversation discusses the existence of a limit along different paths for a given function. The limit along the path x=0 tends to 0, while the limit along y=0 results in an equation with divisibility by zero, indicating that the limit does not exist. The limit along the path y=x^2 tends to 1, but because the limit along y=0 is undefined, it cannot be used to determine the overall existence of the limit. The possibility of using polar coordinates to analyze the limit is also mentioned.
  • #1
ek124
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TL;DR Summary
how to show a multivariable limit does not exist/exist
I want to show that the limit of the following exists or does not exist:

245722

When going along the path x=0 the limit will tend to 0 thus if the limit exists it will be approaching the value 0
when going along the path y=0, we get an equation with divisibility by zero. Since this is not possible does this already show that the limit does not exist? Or does it simply mean that there is an asymptote... I would like to know what this means.
Finally when going along the path y=x^2, the limit tends to 1. Since the first and last path give a different outcome the limit DNE however I want to know what the path along y=0 tells us anyway.
 
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  • #2
nvm it's solved
 
  • #3
i.e. does not exist ?
 
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  • #4
ek124 said:
I want to know what the path along y=0 tells us anyway.

I was going to say if you rewrite the function inside the limit as ## \frac {x^2} {y} + y ##, then perhaps that offers a better perspective but I'm not sure.

I think actually seeing that the top has ##x^2 + y^2##, that should immediately strike you as saying that changing the function into its polar equivalent might be helpful for the limit. We know that ##x^2 + y^2 = r^2## and ##y = r * \sin\theta##, and the limit along the line ##y = 0## corresponds to ## \theta = 0## from the right side and ##\theta = \pi ## from the left, hence we have:

##\lim_{(r, \theta) \rightarrow (0,0)} {\frac {r^2} {r*\sin\theta}} = \lim_{(r, \theta) \rightarrow (0,0)} {\frac {r} {\sin\theta}}##

Yeah this seems to lead to the same undefined issue as well.

Even thinking about this numerically leads to the same issues.

So I think if a function is undefined for that entire line of ##y=0##, then we cannot really use the idea of a limit here. I could be wrong, but that's what I got for now.
 

Related to Showing that a multivariable limit does not exist

1. How do you determine if a multivariable limit does not exist?

To determine if a multivariable limit does not exist, you need to evaluate the limit along different paths approaching the point in question. If the limit values along different paths are not equal, then the limit does not exist.

2. Can a multivariable limit not exist even if the limit values along different paths are equal?

Yes, a multivariable limit can still not exist even if the limit values along different paths are equal. This can happen if the limit value approaches different values as the point is approached from different directions.

3. What is the difference between a multivariable limit not existing and being undefined?

A multivariable limit not existing means that the limit value does not exist at all. On the other hand, a limit being undefined means that the limit value may exist, but it cannot be determined using the given information.

4. Can a multivariable limit not exist at a specific point but exist everywhere else?

Yes, a multivariable limit can not exist at a specific point but exist everywhere else. This can happen if the function is discontinuous at that point or if the limit value approaches different values from different directions.

5. Is it possible for a multivariable limit to not exist at a point but still have a limit at that point?

No, if a multivariable limit does not exist at a point, then it does not have a limit at that point. A limit can only exist if the limit values along all paths approaching the point are equal.

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