MTH 251 – Differential Calculus Chapter 2 – Limits and ContinuityWarnings & ConsiderationsJust-In-Time Algebra ReviewSlide 4The Limit of a FunctionFinding for a given GraphicallyFinding for a given AlgebraicallySlide 8MTH 251 – Differential CalculusChapter 2 – Limits and ContinuitySection 2.3The Precise Definition of a LimitCopyright © 2008 by Ron Wallace, all rights reserved.Warnings & Considerations•Up to this point, there has been no clear definition of a limit.“… it appears to approach …”“… it gets close to …”•Not to mathematical … need a clear definition!•The definition will NOT help to determine the value of a limit, it can only be used to VERIFY that what you think is true is correct (or not).•The definition can also be used to prove the laws of the previous section (details … see appendix 2).•What does the following inequality really mean?•Example …•That is …Just-In-Time Algebra Reviewx a b- <2 5x - <2 5 2 5 2 5 or 5 2 5x xx- < � - < < +- < - < x a b b x a b- < � - < - <Just-In-Time Algebra Review•What does the following inequality really mean?•Example …•That is …0 x a b< - <0 2 5x< - <0 2 5 5 2 5, 2x x x< - < � - < - < �0 , x a b b x a b x a< - < � - < - < �L + L - if, for every number > 0, there exists a number > 0 such that …The Limit of a Functionlim ( )x cf x L�=0 ( )x c f x Ld e< - < � - <Lcc - c + •Find a that satisfies the definition when = 13+1 = 43-1 = 2Finding for a given Graphically131lim 3xx�=31412112d �13Finding for a given Algebraically•Find a that satisfies the definition when = 11( ) 3 1xf x L- = - <1 13 3Need: or x xd d d- < - < - <1 1 1 1 16 12 3 12 6x- <- < - < <1 1 112 3 6x- < - <1 14 2x< <12 4x< <11 3 1x- < - <112d =131lim 3xx�=Finding for a given Algebraically•Find a that satisfies the definition for any > 01( ) 3xf x L e- = - <1 13 3Need: or x xd d d- < - < - <13(3 ) 3 3(3 )xe ee e+ -- < - <1 1 1 1 13 3 3 3 3xe e+ -- < - < -1 13 3xe e+ -< <13 3xe e- < < +13xe e- < - <3(3 )eed+=131lim
View Full Document
Unlocking...