Finding the Limit of a Function as Two Variables Approach the Origin

The limit as X and Y approaches the origin of $\frac{x^2 + y^2}{x+y}$

This problem involves finding the limit of a multivariable function as both variables approach a specific point, which in this case is the origin. Understanding how to solve this type of problem requires knowledge of limits in multivariable calculus, which can be more complex than single variable limits due to the additional dimensions involved. The primary concept to grasp is that the limit must be the same regardless of the path taken to approach the origin. Thus, different methods can be applied to check the consistency of the limit, such as polar coordinates or using different approaches along the coordinate axes.

The denominator in this problem involves a sum of variables, which suggests considering substitutions like polar coordinates where the expression can be simplified using the radius. This transformation can sometimes aid in determining if the limit exists by turning a problem of two variables into one involving just the distance from the origin. The challenge is to demonstrate that as x and y both approach zero via any path — whether linear, parabolic, or others — the outcome remains consistent. If the limit varies with the path, then the overall limit does not exist.

This problem exemplifies the intricacies of multivariable calculus and highlights the importance of understanding how change of variables can simplify the evaluation of limits. The overarching strategy is to dissect the given expression into more manageable components and apply different limit laws and techniques to establish the consistency or inconsistency of the limit.

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