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
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This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.

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

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