# Limits: Evaluating function of 2-variables where a limit doesn't exist

1. Nov 8, 2012

### petertheta

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

Evaluate:

$$\lim_{(x,y)\to(0,0)} \sqrt{x^2+y^2}\sin \frac{1}{\tan xy}$$

**Apologies that this Tex didn't come out, I can't see where the typos are**Hopefully you can still determine the function I am trying to write**

3. The attempt at a solution

So, I can see this isn't solveable by just plugging in the co-ords as the sine function will not be valid. I can't separate them as the Sine function isn't continous due to Tan (xy) being in the denominator. I can't fix x or y or substitute a function for y = g(x).

I can't see a way forward or another example that would give me the direction I'm lacking.

Any pointers guys?

Last edited by a moderator: Nov 8, 2012
2. Nov 8, 2012

### Zondrina

I believe that your function is :

$lim_{(x,y)\to(0,0)} \sqrt{x^2+y^2} sin(\frac{1}{\tan(xy)})$

3. Nov 8, 2012

### petertheta

Yep. Noted the differences. Thanks

4. Nov 8, 2012

### Zondrina

What do you know about the squeeze theorem? What do you know about |sinx| ?

5. Nov 8, 2012

### petertheta

Not heard of it before. Just watched a tutorial online and an interesting theorem. It was only in one variable and don't see much out ther on 2-variable functions. Not sure about the modulus of sin(x)? It's alawys positive?

6. Nov 8, 2012

### SammyS

Staff Emeritus
The function, $\displaystyle f(x,\,y)= \sqrt{x^2+y^2}\sin(\frac{1}{\tan(xy)})$ is a mess near (0, 0).

Perhaps the problem is: find $\displaystyle \lim_{(x,y)\to(0,0)}\sqrt{x^2+y^2}\sin(\tan^{-1}(xy))\ ?$

7. Nov 8, 2012

### petertheta

No the original equation is correct. Definately not arctan!

8. Nov 8, 2012

### Zondrina

Like I said, what do you know about |sin(x)| for any x.

9. Nov 8, 2012

### SammyS

Staff Emeritus
OK.

Putting arctan in there would make the problem uninteresting .

Zondrina has the right approach anyway .

10. Nov 8, 2012

### petertheta

The only thing i can say about it is its always positive... not sure how that helps as it's not the modulus in the equation and1/ tan(xy) will alwys be undefined at (0,0)???

11. Nov 8, 2012

### Zondrina

That's not it. Think about the graph of sin(x). How does it relate to |sin(x)|?

12. Nov 8, 2012

### SammyS

Staff Emeritus
What is the range of sin(x) ?

13. Nov 8, 2012

### petertheta

the range for sin(x) is [-1,1] and for the modulus is [0,1].

Could you be a bit more explicit with where to go with this. How do you deal with the tangent in the denominator with this?

14. Nov 8, 2012

### Zondrina

Exactly, the range of sin(x) is the interval [-1,1]. Now what does this tell you about |sin(x)|??

15. Nov 8, 2012

### petertheta

I'm sorry I just don't know other that its interval is [0,1]...?

16. Nov 8, 2012

### Zondrina

Do you understand that |x| < 1 implies that -1 < x < 1?

Apply this concept to |sin(x)|

17. Nov 8, 2012

### petertheta

Yes, I understand the inequalities you mention but I have not been taught how to manipulate the original question using this method.

Thanks for your help though. I'll try to catch one of my tutors or classmates to help explain.

P.