# Two Var Limit

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1. Feb 9, 2017

### Kaura

1. The problem statement, all variables and given/known data

limit (x -> 0 y -> 0) of xy/sin(x+y)

2. Relevant equations

None that come to mind but maybe Lopital's Rule

3. The attempt at a solution

I know that the limit does not exist but I am having trouble figuring out how to show that it does not

using the line x=y gives x^2/sin(2x)
using the line -x=y gives -x^2/sin(0)

I am not sure what to do from here or if I am just completely missing what to do

2. Feb 9, 2017

### andrewkirk

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.

3. Feb 9, 2017

### Kaura

The limit for x^2/sin(2x) is 0
I cannot think of another line at the moment though

4. Feb 9, 2017

### andrewkirk

What are the simplest lines you might try?

5. Feb 9, 2017

### Kaura

For -x=y it gives -x^2/sin(0) which is undefined I think is there another line that is better though?

6. Feb 9, 2017

### andrewkirk

Those are both sloping lines. Try a non-sloping line.

7. Feb 10, 2017

### haruspex

Job done?

8. Feb 10, 2017

### andrewkirk

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.

9. Feb 10, 2017

### haruspex

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).

10. Feb 10, 2017

### andrewkirk

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.

11. Feb 10, 2017

### haruspex

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.

12. Feb 10, 2017

### andrewkirk

@haruspex: My mistake. You're right. Somehow I'd got it into my head that the numerator was x+y rather than xy.