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Lecture 7

Lecture 7 - 2.2.pdf

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Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

88CHAPTER 2LIMITS AND DERIVATIVESTHE LIMIT OF A FUNCTION22Having seen in the preceding section how limits arise when we want to nd the tangent toa curve or the velocity of an objectwe now turn our attention to limits in general andnumerical and graphical methods for computing them2Lets investigate the behavior of the function dened by for valfxxx2fues of near 2 The following table gives values of for values of close to 2but notxfxxequal to 2yxfxxfx308000000102000000255750000152750000yx2approaches42246400001834400004214310000193710000205415250019538525002014030100199397010020054015025199539850252001400300119993997001x02As x approaches 2From the table and the graph of a parabola shown in Figure 1 we see that when isxfFIGURE 1close to 2 on either side of 2is close to 4 In factit appears that we can make thefxvalues of as close as we like to 4 by taking sufciently close to 2 We express thisxfx2by saying the limit of the function as approaches 2 is equal to 4xfxxx2The notation for this is2lim xx24xl2In generalwe use the following notation1DEFINITIONWe writelim fxLxlaand saythe limit of as approaches equals Laxfxif we can make the values of arbitrarily close to as close to Las we likeLfxby taking xto be sufciently close to on either side ofbut not equal to aaaMATH140Lecture 7 NotesLimit of a FunctionDenition of a limit lim fxL xaRoughly speakingthis says that the values of tend to get closer and closer to thefxnumber as gets closer and closer to the number from either side ofbutxaaaxLIf we can make the values of fx arbitrarily close to L by taking x to be sufciently close to a on either side of a BUT not equal to a This means that the values of fx tend A more precise denition will be given in Section 24to get closer and close to the number L as x gets closer and closer to the number a SECTION 22THE LIMIT OF A FUNCTION89where xaAn alternative notation forNotice the phrase but in the denition of limit This means that in nding thexalimit of as approaches we never considerIn factneed not even befxxaaxfxlim fxLdened whenThe only thing that matters is how is denednearafxaxlaFigure 2 shows the graphs of three functions Note that in part cis not denedfaWe usually use the following notation for nding limitsand in part b But in each caseregardless of what happens at it is true thatafaLlim fxLxlaExample When we are asked to evaluate lim x2 x22 we are asking what SECTION 22THE LIMIT OF A FUNCTION89yyyisasxlafxlLhappens to x22 when x becomes very close to but not equal to 2Notice the phrase but in the denition of limit This means that in nding thexaLLLlimit of as approaches we never considerIn factneed not even befxxaaxfxThe logical answer is that x22 becomes very close but not equal to 2226 which is usually read approaches as approaches axLfxdened whenThe only thi
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