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Unformatted text preview: Lecture 2 : Wednesday April 1st firstname.lastname@example.org Key concepts : Limit, continuity Know how to find limits with - and show they dont exist and use properties of limits 2.1 Definition of limits The definition of a limit for a function f : R n R of several variables is a natural extension of the limit for functions of one variable. The main difference is that instead of saying | x- a | < , we say d ( x,a ) < where we recall that d ( x,a ) = p ( x 1- a 1 ) 2 + ( x 2- a 2 ) 2 + + ( x n- a n ) 2 . In the case n = 1, notice that d ( x,a ) = | x- a | , so this agrees with the one-dimensional definition of the limit. So here is the definition of the limit upon which all our further calculus is based: Definition of the limit The limit of a function f : R n R as x a is L lim x a f ( x ) = L if for every > 0 there exists a > 0 such that if d ( x,a ) < then | f ( x )- L | < . The function f is continuous at a if L = f ( a ). This definition in words says that if x is close to a then f ( x ) should be close to L . Limits of functions of several variables have many of the same properties as the limits we are familiar with for functions of one variable. These properties are listed below here it is assumed that the limits of each of the functions f and g exist and g has a non-zero limit in the third property. Properties of limits [Sum Rule] lim x a ( f ( x ) + g ( x )) = lim x a f ( x ) + lim x a g ( x ) [Product Rule] lim x a ( f ( x ) g ( x )) = lim x a f ( x ) lim x a g ( x ) [Quotient Rule] lim...
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