Solving Multivariable Limits: Evaluating $\lim_{(x,y) \to (0,0)}\frac{x-y}{x+y}$

[spacefreak]

Homework Statement

Evaluate the following limit or give a reason explaining why the limit does not exist.

$$\lim_{(x,y) \to (0,0)}\frac{x-y}{x+y}$$

Homework Equations

$$x = r*\cos\theta$$
$$y = r*\sin\theta$$

The Attempt at a Solution

$$\lim_{r \to 0}\frac{r*\cos\theta-r*\sin\theta}{r*\cos\theta+r*\sin\theta} = \lim_{r \to 0}\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta} = \lim_{r \to 0}\frac{1}{1+\tan\theta} - \lim_{r \to 0}\frac{1}{1+\cot\theta}$$

When I get to this point, I'm stuck. How do I either find the limit or show that it doesn't exist?

 

[Dick]

No need for polar coordinates on this one. Take the limit as x->0 while y=0 and vice versa.

 

[spacefreak]

So, to make sure I understand.

When x -> 0 while y = 0, the limit equals 1. When y -> 0 while x = 0, the limit equals -1. Therefore, the limit does not exist. Am I correct?

I appreciate your help.

 

[Dick]

So, to make sure I understand.

When x -> 0 while y = 0, the limit equals 1. When y -> 0 while x = 0, the limit equals -1. Therefore, the limit does not exist. Am I correct?

I appreciate your help.

Exactly.