[b]1. The problem statement, all variables and given/known data[/b]A

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In summary, the limit as (x,y) approaches (0,0) for (x^4*y^2)/(x^2 + y^2)^2 equals 0 and can be proven using polar coordinates. However, for the limit of x^4 / (y+x^2) as (x,y) approaches (0,0), the existence of a limit is not enough to prove that it equals zero, as shown by the path x=t, y=(-t^2). The parabolic path y=-x^2 must also be considered in order to fully determine the limit.
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Homework Statement


A few:
(1) lim(x,y) -> (0,0) of (x^4*y^2)/(x^2 + y^2)2
(2) lim(x,y) -> (0,0) of x^4 / (y+x^2)

Homework Equations


The Attempt at a Solution


(1) i get that it exists and equals 0 by using polar coordinates
(2) i get that it exists (except for the parabolic path y=-x^2) and equals zero [using (t, mt) and (t, mt^2)]
 
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  • #2


1) looks fine. For 2) how can you have a limit 'except for' something? Look at the path x=t, y=(-t^2). Just because the limit exists on some paths doesn't prove anything.
 
  • #3


yeah but the question says "please ignore the parabolic path y=-x^2" :)
 
  • #4


Since the denominator vanishes on that path, what do you think happens very close to that path?
 

What is the problem statement in "A"?

The problem statement in "A" is a clear and concise explanation of the issue or question that the scientific study aims to address. It outlines the purpose and objectives of the study and provides a context for the research.

What are the variables in "A"?

Variables in "A" refer to any factors or conditions that may affect the outcome of the study. These can be independent variables (those that are manipulated by the researcher) or dependent variables (those that are measured as a result of the independent variables).

What information is given or known in "A"?

The given or known information in "A" refers to any data, facts, or background knowledge that is already established or available before the study begins. This may include previous research findings, existing theories, or specific details about the subject or phenomenon being studied.

Why is it important to clearly define the problem statement and variables in a scientific study?

Defining the problem statement and variables in a scientific study is crucial because it provides a clear focus for the research and helps to avoid any ambiguity or confusion. It also allows other researchers to replicate the study and verify the results, and ensures that any conclusions drawn are based on solid evidence and not influenced by any extraneous factors.

How do scientists go about formulating a problem statement and identifying variables in a study?

Scientists typically use a systematic approach to formulate a problem statement and identify variables in a study. This may involve conducting a literature review, developing a hypothesis, and then determining the key variables that need to be controlled or measured in order to test the hypothesis. The problem statement and variables may also be refined and revised throughout the course of the study as new information is gathered.

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