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# 14.2 - Section 15.2 Limits and Continuity Generalizing...

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Section 15.2 Limits and Continuity “Generalizing Ideas from Single Variable Calculus” The ideas of limits and continuity were critical when defining the der- vative in single variable calculus - f ( x ) = lim h 0 f ( x + h ) f ( x ) h . We shall generalize these ideas to functions of more than one variable. 1. Limits of Functions of Two Variables Recall that the naive idea of the limit of a function f ( x ) at a point x = a is the following: it is the value f ( x ) tends towards as x gets close to a , where by “close”, we mean | x a | is sufficiently small. This causes a problem when defining a limit of a function of two variables - the value | ( x, y ) ( a, b ) | makes no sense. However, the term “close” refers to distance, and we have a general formula for distance between points in the plane. Specifically, two points ( x, y ) and ( a, b ) are close provided radicalbig ( x a ) 2 + ( y b ) 2 is sufficiently small. We can use this definition to define the limit at a point of a function of two variables. Definition 1.1. Let f be a function of two variables whose domain includes points arbitrarily close to ( a, b ) (though not necessarily ( a, b )). Then we say the limit of f ( x, y ) as ( x, y ) approaches ( a, b ) is L and write lim ( x,y ) ( a,b ) f ( x, y ) = L if for every number ε > 0, there exists a number δ > 0 such that | f ( x, y ) L | < ε whenever ( x, y ) is in the domain of f and radicalbig ( x a ) 2 + ( y b ) 2 < δ. Visually, since the domain of f ( x, y ) is a region in 2-space, this means that if we choose a point ( x, y ) within a ball of radius δ > 0 centered at ( a, b ), then the value | f ( a, b ) f ( x, y ) | < ε . The major difference

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14.2 - Section 15.2 Limits and Continuity Generalizing...

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