Multivariable Epsilon-Delta Limit Definitions

For the limit of a multivariable function, consider the two-variable function . (Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) The same limit definition applies here as in the one-variable case, but because the domain of the function is now defined by two variables, distance is measured as , all pairs within of are considered, and should be within of for all such pairs .
As an example, here is a proof that the limit of is 10 as . Claim: for a given , choosing satisfies the appropriate conditions for the definition of a limit: (the given condition) reduces to , which implies that and .

Now, by the triangle inequality, and . If , , and if , . Thus by the choice of , , and because is arbitrary, an appropriate can be found for any value of ; hence the limit is 10.