Nonexistence of a Multivariable Limit

In multivariable calculus, a limit of a function exists at a point if and only if we can make as close as we want to for all points arbitrarily close to One way to show that a limit does not exist (i.e. the definition fails) is to show that the function approaches different values from different directions. Akin to the notion of a one-sided limit in single-variable calculus, we consider the limit along a path in multivariable calculus. If the limit exists and is the same along all paths to then the limit of the function exists.
This Demonstration gives an example where the limit does not exist. It examines the function , which is defined for all values of in except the origin. We consider the limit of the function as we approach the origin along the paths for all values of between and . Notice that the function approaches as approaches the origin along the path , yet the function approaches as approaches the origin along the path As the limits along various paths do not agree, the limit of the function does not exist.

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