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2224-Sec12_2-HWT - Mat h 22 24 Multiva ri a ble C alc Sec...

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Math 2224 Multivariable Calc – Sec. 12.2: Limits and Continuity in Higher Dimensions I. Review from math 1205 A. Limits 1. Def n : Let f be a fn defined on some open interval that contains the number a , except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L , and we write lim x ! a f x ( ) = L if for every number ε >0 there is a corresponding number δ >0 such that | f(x)-L |< ε whenever 0<| x-a |< δ Another way of writing the last line of this def n is: if 0<| x-a |< δ then | f(x)-L |< ε 2. Th m : lim x ! a f x ( ) =L iff lim x ! a - f x ( ) = lim x ! a + f x ( ) =L, where a , L are real numbers. 3. Some of the Limit laws Let f and g be any 2 functions st lim x ! a f x ( ) = L and lim x ! a g x ( ) = M and suppose that c is a constant. Then a. Sum/Difference Rule: lim x ! a f x ( ) ± g x ( ) " # $ % = lim x ! a f x ( ) ± lim x ! a g x ( ) = L ± M b. Constant Multiple: lim x ! a c f x ( ) " # $ % = c lim x ! a f x ( ) = cL c. Product Rule: lim x ! a f x ( ) g x ( ) " # $ % = lim x ! a f x ( ) & lim x ! a g x ( ) d. Quotient Rule: lim x ! a f x ( ) g x ( ) " # $ % & = lim x ! a f x ( ) lim x ! a g x ( ) , provided lim x ! a g x ( ) exists e. Power Rule: If n
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