Math 314 Lecture #12 14.2: Limits and Continuity

Math 314 Lecture #12 : Limits and ContinuityOutcome A: Recall and apply the definition of limit of a function of several a function of two variables whose domainDcontains points arbitrarily close tothe point (a,b).We say thelimitoff(x,y) as (x,y) approaches (a,b) (within the domainD) is thenumberLand we writeL= lim(x,y) (a,b)f(x,y),if for every number >0 there exists a corresponding number >0 such that for allpoints (x,y) Dwithin a distance of from (a,b) there holds|f(x,y) L|< .There are infinitely many ways for a point (x,y) to approach (a,b): straight line ap-proaches, quadratic and cubic approaches, squiggly approaches, spiral approaches, are illustrated for (x,y) approaching (0,0).When the limitLexists forf(x,y) as (x,y) approaches (a,b), EVERY approach of (x,y)towards (a,b) gives the same limiting value the other hand, when there are two different approaches of (x,y) towards (a,b) thatgive different limiting values off, then the limit offas (x,y) approaches (a,b)does the limit, if it exists, or show that the limit does not exist.

§14.2: Limits and Continuity Outcome A: Recall and apply the definition of limit of a function of several variables. Let f be a function of two variables whose domain D contains points arbitrarily close to the point (a,b). We say the limit of f(x,y) as (x,y) approaches (a,b) (within the domain D) is the number L and we write L = lim (x,y ...

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  Continuity, Limits, Limits and continuity

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