How to prove the existence of a limit for a function with two variables?

[Telemachus]

Hi there. Well, in the next exercise I must find the limit of $$\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{\cos(xy)-1}{x}}$$, if it exists. I want to know if I did it right.

If $$y=cx$$

$$\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{\cos(xy)-1}{x}}=\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{\cos(cx^2)-1}{x}}=\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{-\sin(cx^2)2cx-1}{1}}=-1$$

If $$x=cy$$

$$\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{\cos(cy^2)-1}{cy}}=\displaystyle\lim_{(x,y) \to{(0,0)}}{\displaystyle\frac{-\sin(cy^2)2cy-1}{cy}}=-\displaystyle\frac{1}{c}$$

$$\therefore{\not{\exists}}\textsf{double limit}$$

So, what you say?

By there, and thanks for posting.

PD: Ok, Now I see, after plotting with wolfram, some calculus errors I've committed. The limit actually seems exists. So I should use the delta epsilon definition of limits to make a demonstration.

 

[Office_Shredder]

If y=cx, then x=dy where d=1/c. You shouldn't get different answers trying each variety.

It looks like you're trying L'hopital's rule, but you aren't differentiating properly (the derivative of 1 is zero for example, and the derivative of cy is not going to be cy). Furthermore you're only differentiating with respect to one variablei

Try expanding cos(xy) in terms of a power series

 

[Telemachus]

You mean like using Taylor? I don't know how to with two variables. I see I've committed a few mistakes. As you say, it should give the same limit, which is zero. But, to demonstrate that zero is the limit I must use the delta epsilon definition, right?