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.

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... 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 ...