Two difficult limit problems

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In summary, the first question is asking if it is possible to determine the value of u when given a limit in terms of x and y with an unknown u, and the second question is asking if it is possible to prove the existence of a limit using the δ and ε method, where n is a dummy variable and L corresponds to a line. Both questions require more clarification and conditions in order to be properly answered.
  • #1
Brad_Ad23
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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!
 
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  • #2
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.
 
  • #3
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..
 

What is a limit in mathematics?

A limit is a fundamental concept in mathematics that describes the behavior of a function as its input approaches a certain value or point. It represents the value that a function tends towards but may never actually reach.

What are the two types of difficult limit problems?

The two types of difficult limit problems are indeterminate forms and oscillating functions. Indeterminate forms occur when the limit of a function cannot be determined through direct substitution. Oscillating functions have values that constantly fluctuate and do not approach a specific value as the input approaches a certain point.

How do you solve an indeterminate form limit problem?

To solve an indeterminate form limit problem, you must manipulate the function algebraically to find a form that can be evaluated. This can involve factoring, simplifying, or using algebraic identities. Once the function is in a solvable form, you can substitute the limit value into the function to find the limit.

What is meant by a limit that does not exist?

A limit that does not exist means that the function does not approach a specific value as the input approaches a certain point. This could be because the function has a discontinuity at that point or because the limit is infinite.

How can you determine if a limit exists?

A limit exists if the values of the function get closer and closer to a specific value as the input approaches a certain point. This can be determined through direct substitution, algebraic manipulation, or by using theorems such as the Squeeze theorem. If the limit does not exist, it means that the function does not approach a specific value and may have a discontinuity or be oscillating.

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