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14.2 - 14.2 LIMITS AND CONTINUITY KIAM HEONG KWA 1 Limits...

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14.2: LIMITS AND CONTINUITY KIAM HEONG KWA 1. Limits of Functions Let f be a function of two variables whose domain D includes points arbitrarily close to ( a, b ). We say that the limit of f ( x, y ) as ( x, y ) approaches ( a, b ) is the number L and we write lim ( x,y ) ( a,b ) f ( x, y ) = L , lim x a, y b f ( x, y ) = L , or f ( x, y ) L as ( x, y ) ( a, b ) if for every > 0 there is a corresponding δ = δ ( ) > 0 such that | f ( x, y ) - L | < whenever ( x, y ) D and 0 < p ( x - a ) 2 + ( y - b ) 2 < δ. Geometrically, this means that if we restrict ( x, y ) to the interior of the disk centered at ( a, b ) and of radius δ , then the corresponding part of the graph of f lies strictly between the planes z = L - and z = L + . See figure 2 on p. 871 of the text for a graphical illustration. Limit Theorems. Let f and g be functions of two variables such that f ( x, y ) L and g ( x, y ) M as ( x, y ) ( a, b ), and let h be a function of one variable. Then lim ( x,y ) ( a,b ) [ f ( x, y ) + g ( x, y )] = L + M ; lim ( x,y ) ( a,b ) [ f ( x, y ) - g ( x, y )] = L - M ; lim ( x,y ) ( a,b ) f ( x, y ) g ( x, y ) = LM ; lim ( x,y ) ( a,b ) f ( x, y ) g ( x, y ) = L M
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