Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as (x, y

step 1 So in this question, we're considering the function F of X, Y, equals the natural log of X squar

Discuss the continuity of the function and evaluate the...

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Key Concepts

Domain of Logarithms

The natural logarithm function is defined only for positive real numbers. In multivariable contexts, when the function is given as ln(s) where s is an expression involving multiple variables, it is necessary to ensure that the expression remains positive. For ln(x²+y²), the argument x²+y² must be greater than zero, which restricts the domain of the function.

Continuity of Multivariable Functions

A function of several variables is continuous at a point if it is defined in a neighborhood of that point and the limit as the point is approached equals the function's value there. In the case of ln(x²+y²), the function is continuous on its domain (where x²+y² > 0), but it is not defined at the point (0,0).

Evaluating Multivariable Limits

When evaluating limits in several dimensions, especially for functions that involve symmetries or radial expressions, it is often useful to employ polar coordinates. Here, converting x²+y² to r² simplifies the expression to ln(r²) which equals 2 ln(r). As r approaches 0, ln(r) tends to negative infinity, indicating that the function does not have a finite limit at (0,0).

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