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mthsc206-fall-2010-notes-15.2

mthsc206-fall-2010-notes-15.2 - 15.2 Limits and Continuity...

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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 δ so that whenever x is a number with 0 < | x - a | < δ 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 didn’t equal, we were able to conclude the two-sided limit didn’t 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 > 0 . 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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