The discussion revolves around evaluating the limit of the expression xy/sin(x+y) as (x,y) approaches (0,0). Participants are exploring the behavior of the limit along various paths and questioning the existence of the limit based on different approaches.
The discussion is active, with participants providing insights and questioning the validity of certain approaches. Some guidance has been offered regarding the use of different lines to approach the limit, and there is an acknowledgment of the need for a non-sloping line to further investigate the limit's behavior.
There is a mention of the assumption that the domain of the function is where it is well-defined, which raises questions about the validity of certain lines of approach. Participants are also considering the symmetry in the problem and the implications of different values of 'a' in the context of lines represented by y=ax.
What limit do you get, approaching the origin along that line?Kaura said:using the line x=y gives x^2/sin(2x)
andrewkirk said:What limit do you get, approaching the origin along that line?
Now, can you think of a different line along which to approach the origin that gives you a different limit. If you can, you will have proven that the general limit does not exist - because if it did, the limits for all different lines of approach would be the same.
L'Hopital's rule will be useful in finding the limit for both lines.
What are the simplest lines you might try?Kaura said:The limit for x^2/sin(2x) is 0
I cannot think of another line at the moment though
Kaura said:figuring out how to show that it does not
Job done?Kaura said:using the line -x=y gives -x^2/sin(0)
Not if one follows the convention that, where the domain of a function is not explicitly stated, it is assumed to be the set of points on which the given formula has a well-defined value. In that case the whole line -x=y is excluded from the domain.haruspex said:Job done?
Ok, but it still provides the basis for the proof of nonconvergence. Sloping straight lines won't do it. We need a curve which is asymptotically y=-x at (0,0).andrewkirk said:Not if one follows the convention that, where the domain of a function is not explicitly stated, it is assumed to be the set of points on which the given formula has a well-defined value. In that case the whole line -x=y is excluded from the domain.
Given the symmetry between x and y, any line can be represented by y=ax. The function is ax2/sin(x(1+a)). If a is not -1 then this clearly tends to 0 as x tends to 0.andrewkirk said:Sloping straight lines won't do it but, if my calcs are correct, a non-sloping line will, given the observation the OP has already made about the limit along the line y=x.