I have searched the forum for a while looking for general help with the epsilon-delta definition of limits for multivariable equations but was unable to find a good reference. I even looked at the sticky post above but it only covered single variable examples. I understand the concept I am just having a hard time figuring out how to get my left hand side to look like a multiple of delta? Can someone can post a link which may help clarify epsilon-delta limit proofs for several variables.

In particular,

(x,y)-->(9,-5) X^5 + 8(X^3)Y - 2XY^2

Since F(x,y) is a polynomial, it is continous therefore I can find the limit by substition L=64,431. What is the epsilon delta method to arrive at this answer.

If I am give epsilon= 1/1000 how do I find delta? I am pretty sure that is how this concept will show up on my exams.