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Chap14_Sec2 - PARTIAL DERIVATIVES 14.2 Limits and...

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PARTIAL DERIVATIVES 14.2 Limits and Continuity In this section, we will learn about: Limits and continuity of various types of functions.
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LIMITS AND CONTINUITY Let f be a function of two variables whose domain D includes points arbitrarily close to ( a , b ). indicates that: square4 The values of f ( x , y ) approach the number L as the point ( x , y ) approaches the point ( a , b ) along any path that stays within the domain of f . ( , ) ( , ) lim ( , ) x y a b f x y L = square4 i.e., for every number ε > 0, there is a corresponding number δ > 0 such that, if then 2 2 ( , ) and 0 ( ) ( ) x y D x a y b δ < - + - < | ( , ) | f x y L ε - < Notice that: square4 is the distance between the numbers f ( x , y ) and L square4 is the distance between the point ( x , y ) and the point ( a , b ). This means that the distance between f ( x , y ) and L can be made arbitrarily small by making the distance from ( x , y ) to ( a , b ) sufficiently small (but not 0). | ( , ) | f x y L - 2 2 ( ) ( ) x a y b - + -
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LIMIT OF A FUNCTION Another illustration of the definition is given here, where the surface S is the graph of f .
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