In Exercises 18 through 21, prove that the function is...

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=\frac{x^{3} y^{2}}{x^{6}+y^{4}}
$$

Key Concepts

Techniques for Analyzing Limits

Common methods such as evaluating the limit along specific paths, including linear paths or non-linear paths (or using polar coordinates), are crucial in determining the existence and nature of limits in multivariable functions. These techniques help in identifying whether or not a discontinuity is removable.

Removable Discontinuity

A removable discontinuity occurs when the limit of the function at a given point exists, but the function is either not defined at that point or is defined differently from the limit. In such cases, the discontinuity can be 'removed' by redefining the function's value to match the limit.

Essential (Non-removable) Discontinuity

An essential discontinuity is characterized by the failure of the limit to exist at the point of discontinuity. Unlike removable discontinuities, these cannot be fixed simply by redefining the function at that point.

Multivariable Limits

In the context of functions of several variables, the limit involves approaching a point along every possible path. For a function to be continuous at a point, the limit as the variables approach that point must not depend on the path taken and must equal the function’s value at that point.

Path Dependency

When evaluating limits in several dimensions, it is common to test the limit along different paths. If the function gives different limit values along different paths, it indicates that the overall limit does not exist at that point, thereby establishing discontinuity.

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