Two difficult limit problems!

The first one is a mere question. If we are given lim f(x,y) = L, (x,y) -->(∞, u) where u represents an unknown, is it possible to figure out what value u must be so that the limit will in fact, be L?


And the second one:

given:

lim [(xx/(x-n)(x-n))-n - ((x-n)(x-n))-n]/n = L for (x,n)--->(∞, 0). Is it possible to prove that this limit exists using the δ and ε method? In this case n is merely a dummy variable, and one can view this as y = f(x,n) if you wish with the value of L corresponding to the line y = L.


Thanks!
 

mathwonk

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These questions are hard for me to understand. In the second one for example, you are given that the limit exists and also asking whether it exists. It seems to me that if a limit exists then yes it must be possible to prove it. Are you really asking whether someone can exhibit a proof that this limit exists, without assuming it exists in advance?

I am having trouble even understanding what the expression is that you are asking about.

E.g. I am confused by your use of the letter n, since that usually denotes an integer. Do you mean for n to take on real non integer values? If not, it makes little sense for it to converge to zero.

In the first one there is not enough information to respond. But if you were given an explicit function then the answer could be yes.

I suggest you ask for more explanation from your source for this problem.
 

arildno

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Assuming the limiting processes in x and y are interchangeable (for simplicity), is your first question:

Find u when given:
[tex]\lim_{y\to{u}}\lim_{x\to\infty}f(x,y)=L[/tex]?

Let's define:
[tex]\lim_{x\to\infty}f(x,y)=G(y)[/tex]

Assuming G(y) continuous, we'll have, if G is invertible, and L in the range of G:
[tex]u=G^{-1}(L}[/tex]

But, as we see there are lots of if's and conditions we must add here in order to gain a properly stated question..
 

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