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

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 - < then ( 29 f x L - < . 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online