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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 29

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 29. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 7.2 Left- and Right-Handed Limits Sometimes the Approach Is Critical 261 limx.0 x limx.0 x 0 limx.0- x limx.0- -x 0 lnm. . x 0. EXAMPLE 7.13 Let x x . a Sketch the graph of and sketch the graph of . b Evaluate 1 and -2 . c Verify that 0 is undefined. SOLUTION a i. graph of x x ii. graph of x jX x x Figure 7.19 b For x 0 x x so 1 1. For x 0 x -x so -2 -1. c Approaching the problem from a graphical viewpoint we notice that the graph of does not appear locally linear at x 0. No matter how much the region around x 0 is magnified the graph has a sharp corner there so it will never look like a straight line. The slope of is -1 if x approaches zero from the left and 1 if x approaches zero from the right. Analytically we can argue similarly. 0 A - 0 0 liim------I------- A.0 A A - 0 lim A.0 A A lim A.0 A This is the limit we looked at in Example 7.11. We concluded that limA.0 does not exist. Therefore 4 x 11 the derivative of x evaluated at x 0 does not exist. jx x 0 EXAMPLE 7.14 Find limf.0 . SOLUTION limi.0 1 likewise limz.0- J2 rc . 262 CHAPTER 7 The Theoretical Backbone Limits and Continuity Therefore we say limz 0 1 - Figure 7.20 The relationship between left- and right-hand limits and two-sided limits can be stated most succinctly as follows. lim x L if and only if lim x lim x L. x a x x The limit as x approaches a of x is L if and only if the left- and right-hand limits are both equal to L. In this section we ve seen that if limx a- x L1 and limx a x L2 but Li L2 then limx a x does not exist. This is not the only situation in which a limit does not exist. Consider for example the function g x that is 1 when x is rational and -1 when x is irrational. limx 0 g x does not exist since in any open interval around zero no matter how small there will be both rational and irrational numbers. In this case neither the left- nor right-hand limit exists. Similarly limx OT g x does not exist because for any N no matter how big N is there will always be both rational and .