Limits of Functions with several variables

[mit_hacker]

Homework Statement

(Q) By considering different paths of approach, show that the limit of the following function does not exist:

lim┬((x,y)→(0,0))⁡〖y/(x^2-y)〗

Homework Equations

y=kx^2 substitution.

The Attempt at a Solution

After substituting, the functions becomes k/(1-k^2). thus, when we consider different paths of approach, (i.e.) when k takes different values, the value of the limit will be different and hence, the limit does not exist.

Can someone please tell me if I'm doing it right? Thanks a ton!

 

[Hootenanny]

Why do you need so use a substitution? Have you quoted the whole question? If the question is as quoted the solution is trivial and there is no need for any substitution just take limits along the two axes.

 

[Math Jeans]

Correct. If you make two different substitutions in which the values of x and y still go to 0, and the limit results in two different values, then the limit does not exist.

 

[mit_hacker]

Thanks a ton!

Thank-you very much! I thought so too but was not sure. Thanks a lot for re-assuring me!