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Find the limit as ( (x, y) ) goes to ( (0,0) ) of ( f(x, y) = frac5xy² + 4y³ + 3x³y ) along the indicated path. a) ( x = y ) b) ( y = 2x² )

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Final answer:

The question is about finding the limit of a multivariable function at a point following a given path in Calculus. The idea is to substitute for x in the function using the given equations, and then approach the limit. However, the limit might not be the same for all paths.

Step-by-step explanation:

The subject of the question is about finding the limit of a function as (x, y) goes to (0,0) along a certain path, which lies within the realm of calculus and specifically in multivariable calculus involving limits and continuity. This involves understanding the concept of path-dependent limits.

In this case, you'll first substitute for x in the function f(x, y) = frac5xy² + 4y³ + 3x³y by using a) x = y, and b) y = 2x², then approach the limit as x goes to 0. Evaluating these limits can sometimes be tricky, as the limit may exist along some paths but not others.

This problem is a good example of limits behaving oddly in multiple dimensions. While a limit might exist along one path towards a point in space, there may be other paths along which the function behaves differently. Hence, a function is only continuous at a point if it gives the same value, no matter which path you approach along.

Learn more about Multivariable Calculus:

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