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**Unformatted text preview: **15.2 Limits and Continuity Recall: A real-valued function f of one variable has a limit L as x approaches the number a , i.e., lim x a f ( x ) = L , if there is an open interval containing a where f is defined (except possibly at x = a ), and if for every positive number there is a positive number so that whenever x is a number with < | x- a | < then | f ( x )- L | < . NOTE: The only way in which to approach the value a was along the x-axis from the left or from the right. So if lim x a- f ( x ) = lim x a + f ( x ) = L , then the two-sided limit existed and was equal to L . On the other hand, if the two one-sided limits didnt equal, we were able to conclude the two-sided limit didnt exist. How do these ideas change when f is a real-valued function of two or more variables? Definition 15.2.1 Let ( a, b ) be a point in the xy-plane and let r > . The open disk centered at ( a, b ) with radius r is denoted D r ( a, b ) and is given by D r ( a, b ) = { ( x, y ) | ( x- a ) 2 + ( y- b ) 2 < r } ....

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