Math 31A 2009 10 01 MATH 31A DISCUSSION JED YANG 1 Basic Limits 1 1 Basic Limit Laws Assume that limx c f x and limx c g x exist Then a Sum Law lim f x g x lim f x lim g x x c x c x c b Constant Multiple Law For any number k R lim kf x k lim f x x c x c c Product Law lim f x g x lim f x lim g x x c x c x c d Quotient Law If limx c g x 6 0 then lim x c limx c f x f x g x limx c g x 1 2 Exercise 2 3 22 Evaluate the limit limz 1 z 1 z z 1 Solution Recall that limz 1 z 1 and limz 1 1 1 By the Quotient Law 1 1 z 1 1 limz 1 z 1 lim z limz 1 z 1 limz 1 z 1 1 By the Sum Law limz 1 z limz 1 z 1 1 2 By the Sum Law limz 1 z 1 2 So by the Quotient Law 1 1 z 2 z z 1 z lim limz 1 z z 1 limz 1 z 1 2 1 1 3 Exercise 2 3 29 Can the Quotient Law be applied to evaluate limx 0 sin x x Solution The Quotient Law requires the limit of the denominator namely limx 0 x to exist and be nonzero This is not the case so we cannot apply directly 1 4 Exercise 2 3 30 Show that the Product Law cannot be used to evaluate limx 2 x 2 tan x Solution The Product Law requires the limit of each factor to exist However limx 2 tan x does not exist 1 5 Exercise 2 3 31 Give an example where limx 0 f x g x exists but neither limx 0 f x nor limx 0 g x exists Solution Let f x be any function defined on a neighborhood of 0 but not necessarily at 0 such that limx 0 f x does not exist e g f x 1 x Let g x f x Then of course limx 0 g x also does not exist otherwise by the Constant Multiple Law limx 0 f x also exists But notice f x g x is identicaly zero in a neighborhood of 0 but not necessarily at 0 So limx 0 f x g x 0 exists Math 31A Yang 2 1 6 Exercise 2 3 32 Assume that the limit La limx 0 a x 1 exists and that limx 0 ax 1 for all a 0 Prove that Lab La Lb for a b 0 Hint ab x 1 ax bx 1 ax 1 x Solution By definition Lab limx 0 ab x 1 limx 0 ax b x 1 a x 1 Since x limx 0 ax 1 by assumption and limx 0 b x 1 Lb exists by assumption the x x Product Law states limx 0 ax b x 1 1 Lb Now limx 0 a x 1 La by assumption x x so the Sum Law yields limx 0 ax b x 1 a x 1 Lb La x 1 7 Exercise 2 3 38 Assuming that limx 0 statements is necessarily true a f 0 0 b limx 0 f x 0 x f x x x 1 which of the following Solution Remember that the value of f x at x 0 never matters when we evaluate the limit limx 0 f x So a is not necessarily true Recall that limx 0 x 0 so by the Product Law limx 0 f x limx 0 x f x x f x f x limx 0 x limx 0 x 0 1 0 Since limx 0 x 1 and limx 0 x 0 we get limx 0 f x 0