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lecture #13. Ch.14.2. Limits and continuity.

# lecture #13. Ch.14.2. Limits and continuity. - Math 234 SS...

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Math 234 SS 2008 CH 14.2. Limits and Continuity in Higher dimensions. Lecture #13 Review (functions of one variable). Limits and Continuity. We begin with a review of the concepts of limits and continuity for real-valued functions of one variable. Definition 1. Let : . f D R R and let a R ̮� Then ( 29 lim x a f x L D = means that for 0 ε 2200 0 δ 5 such that if , x a δ - < then ( 29 f x L ε - < . From the definition easily follow two fundamental results about the limits. 1). If c R D , then lim x a c c D = 2) lim x a x a for a R D = 2200 The basic facts used to compute limits are contained in the following theorem. Theorem 1. Basic Limits Theorem. Let , : . f g D R R ̮ Suppose ( 29 ( 29 lim lim x a x a f x L and g x M D = = . Then ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 1.lim 2. lim 3.lim 0. x a x a x a f x g x L M f x g x L M f x L provided M g x M D D D + = + = = 1

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Math 234 SS 2008 CH 14.2. Limits and Continuity in Higher dimensions. Lecture #13 Theorem 2. The Squeeze (Sandwich)Theorem . If ( 29 ( 29 lim lim x a x a f x L g x D = = and if ( 29 ( 29 ( 29 f x h x g x D , then ( 29 lim . x a h x L D = From these assertions follows that for any polynomial or rational function (recall that a rational function is a quotient of two polynomials) ( 29 ( 29 lim x a Q x Q a D = for Q a D 2200 . Continuity was defined taking a hint from above result. Definition 2. Let : . f D R R and let a D ̮� Then f is continuous at a means ( 29 ( 29 lim . x a f x f a D = By the comments preceding Definition 2 each rational function is continuous at each point in its domain. The same is true for all of
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lecture #13. Ch.14.2. Limits and continuity. - Math 234 SS...

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