Evaluating a Limit: Is this Right or Wrong?

[Odyssey]

Hi all,

Can you tell me if the method of evaluating this limit is right or wrong please?

The limit is:
$$\lim_{\substack{x\rightarrow 0\\y\rightarrow 0\\z\rightarrow 0}} f(x,y)=\frac{x^2+y^2-z^2}{x^2+y^2+z^2}$$

I evaluate it along the x-axis, y-axis, and z-axis...
$$\lim_{\substack{x\rightarrow 0\\y=0\\z=0}} f(x,y)=\frac{x^2+y^2-z^2}{x^2+y^2+z^2}=\frac{x^2}{x^2}=1$$
and similarly the limit along the y-axis is 1, and the limit along the z-axis is -1.

Since the limits do not equal, the limit DNE.

Is this right or is this wrong? Thank you for the help!

 

[arildno]

That's correct.

 

[Odyssey]

Thank you for the help! =D

 

[stunner5000pt]

That's correct.

i thought you had to use polar co ordinates or the epsilon way of proving the limit rather than merely holding 2 constant and plugging one in?

 

[Hurkyl]

He's not trying to prove the limit exists... he's proving it doesn't exist.

 

[arildno]

i thought you had to use polar co ordinates or the epsilon way of proving the limit rather than merely holding 2 constant and plugging one in?

That is roughly what you need to do in order to prove that the limit DOES exist.
(In general, damn hard)
However, if you can show that along two different paths towards your point, the limiting value is different, then you have proven the limit CANNOT exist.
(Remember, the limit must be the same along every imaginable path, in order to exist!)