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Unformatted text preview: Section 14.2 Limits and Continuity in Higher Dimensions Shubhodeep Mukherji 1 Limit of a Function of Two Variables O We say that a function f ( x,y ) approaches the limit L as ( x,y ) approaches ( x ,y ), and write lim ( x,y ) → ( x ,y ) f ( x,y ) = L if, for every number > 0, there exists a corresponding number δ > 0 such that for all ( x,y ) in the domain of f , | f ( x,y )- L < | whenever < p ( x- x ) 2 + ( y- y ) 2 < δ. This implies that the distance between f ( x,y ) and L becomes ar- bitrarily small when the distance from ( x,y ) and ( x ,y ) becomes sufficiently small. ( x,y ) must remain in the domain at all times. 1.1 Properties of Limits of Functions of Two Variables 1. Sum/Difference Rule: lim ( x,y ) → ( x ,y ) ( f ( x,y ) ± g ( x,y )) = L ± M 2. Product Rule: lim ( x,y ) → ( x ,y ) ( f ( x,y ) g ( x,y )) = LM 3. Constant Multiple Rule: lim ( x,y ) → ( x ,y ) ( kf ( x,y )) = kL 4. Quotient Rule: lim ( x,y ) → ( x ,y ) f ( x,y ) g ( x,y ) = L M 1 5. Power Rule: If5....
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