We examine the above function and consider its limit as (x,y)-> (0,0).We take two different paths in the x-y plane for approaching the point (0,0),and find that f(x,y) approaches two different values .This enables us to conclude that the given limit does not exist.For a detailed discussion see the solution given.© BrainMass Inc. brainmass.com September 20, 2018, 3:46 am ad1c9bdddf - https://brainmass.com/math/calculus-and-analysis/establish-non-existence-multivariable-limit-245416
Find limit of the function x^2y^2/[x^2y^2 + (x - y)^2] as (x,y) -> (0,0).
To explain the procedure, first let us consider the case of a single variable function by examining the limit of the function f(x) = |x|/x as x->0. If we are able to show that the left hand limit and right hand limits are unequal then the limit does not exist.
When we consider the left hand limit then x->0 by taking values less than zero that is by taking negative values. When x < 0,then |x| = -x and accordingly |x|/x has value -1; this means that ...
We consider the limit of a given function f(x,y) of two variables x and y as the point (x,y) approaches a given point (a,b). By taking two different paths for approaching (a,b) ,we show that f(x,y) approaches two different values.This allows us to conclude that the limit in question does not exist.