To establish non-existence of multivariable limit
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.© SolutionLibrary Inc. solutionlibary.com 9836dcf9d7 https://solutionlibrary.com/math/calculus-and-analysis/to-establish-non-existence-of-multivariable-limit-7fl8
... left hand limit of f(x) =|x|/x is -1.
Now we consider the right hand limit of f(x). In this case,x->0 by remaining positive. So here |x| = x and therefore |x|/x has value + 1, which means that the right hand limit of f(x) = |x|/x,as x ->0,is +1.
Since the left hand limit and right hand limits are unequal, the limit of f(x) = |x|/x,as x->0, does not exist. We thus see that when we approach 0 through two different paths and we get two different values, the limit does not exist. Similar is the situation with functions of ...