tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 28

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 28. 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 | Investigating Limits Methods of Inquiry and a Definition 251 beginning of this section we will not be applying this definition much in practice it is the meaning behind the definition that is of primary importance to us. f For all x within 3 of 3 f x is well within e of L. i For all x within 3 of 3 f x is within e of L. ii Figure EXAMPLE Consider the function x 2 for x 3 undefined for x 3. We can write this function compactly as f x _3 because x x_3 x for x 3 and is undefined at x 3. Find limx 3 X X_f SOLUTION EXAMPLE x x 3 x lim -------- lim - provided x 3 x--3 2 x _ 3 x .3 2 _ 3 2 The argument given in Example holds without alteration since we always worked with the condition x 3. A single hole in the graph makes no difference in the limit. In fact inserting any finite number of holes in the graph of a function has no effect on the computation of limits. Let f x 2 for x 3 2 for x 3. Find limx 3 f x . 252 CHAPTER 7 The Theoretical Backbone Limits and Continuity SOLUTION Again x 2 since x 2 for x 3 and the arguments given above both formal and informal hold here as well. Notice that we ve established that x 3 for each of the functions x drawn below. The limit tells us about the behavior of near x 3 but not at x 3. EXAMPLE Find 4xxx2 We re interested in the behavior of x 4x x2 4 x x for x near zero but not at x 0. For x 0 L 4 x so liim .o L lim x . 4 x 4. More formally we must show that for any positive e x - 4 e provided x is small enough and nonzero . For x 0 we have l x - 4 4 x x - 4 4 x - 4 so if 0 x e then x - 4 e as well. 4 x for x 0 undefined for x 0 The limit in Example is the limit that we would compute if we were computing the derivative of g x x2 at x 2. Investigating Limits Methods of Inquiry and a Definition 253 2 h - g 2 g 2 lim ------------- h û h 2 h 2 - 4 lim ----------- h û h 4 4h h2 4 lim ------------- h û h lim ----- This is the limit given in Example . h o h lim 4 h hû EXERCISE Show .