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print barcode image c# Copyright 2003 by The McGrawHill Companies, Inc Click Here for Terms of Use in Software
Copyright 2003 by The McGrawHill Companies, Inc Click Here for Terms of Use Making Quick Response Code In None Using Barcode creation for Software Control to generate, create QR Code JIS X 0510 image in Software applications. Decoding Quick Response Code In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Foundations of Calculus
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Draw Code39 In None Using Barcode maker for Software Control to generate, create Code39 image in Software applications. GS1128 Encoder In None Using Barcode maker for Software Control to generate, create EAN 128 image in Software applications. See Fig 21 Calculate limx 1 f (x) Code 128C Drawer In None Using Barcode encoder for Software Control to generate, create Code128 image in Software applications. Bar Code Encoder In None Using Barcode generator for Software Control to generate, create bar code image in Software applications. SOLUTION Observe that, when x is to the left of 1 and very near to 1 then f (x) = 3 x is very near to 2 Likewise, when x is to the right of 1 and very near to 1 then f (x) = x 2 + 1 is very near to 2 We conclude that USS ITF 2/5 Maker In None Using Barcode printer for Software Control to generate, create ANSI/AIM ITF 25 image in Software applications. Making GS1 DataBar Expanded In Java Using Barcode generation for Java Control to generate, create GS1 DataBar image in Java applications. We have successfully calculated our rst limit Figure 21 con rms the conclusion that our calculations derived EXAMPLE 22 UCC.EAN  128 Drawer In Visual C#.NET Using Barcode creation for VS .NET Control to generate, create GTIN  128 image in VS .NET applications. Encode Bar Code In None Using Barcode generation for Microsoft Word Control to generate, create barcode image in Office Word applications. De ne g(x) = Calculate limx 2 g(x) x2 4 x 2
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Fig 21
lim f (x) = 2 CHAPTER 2 Foundations of Calculus
calculate limx 2 ), we can then divide the denominator of the expression de ning g into the numerator Thus g(x) = x + 2 Now for x = 2
lim g(x) = lim x + 2 = 4
Fig 22
The graph of the function g is shown in Fig 22 We encourage the reader to use a pocket calculator to calculate values of g for x near 2 but unequal to 2 to check the validity of our answer For example, x 18 19 199 1999 2001 201 21 22 g(x) = [x 2 4]/[x 2] 38 39 399 3999 4001 401 41 42 We see that, when x is close to 2 (but unequal to 2), then g(x) is close (indeed, as close as we please) to 4 You Try It: Calculate the limit limx 3 x 3 3x 2 + x 3 x 3 Math Note: It must be stressed that, when we calculate limx c f (x), we do not evaluate f at c In the last example it would have been impossible to do so We want to determine what we anticipate f will do as x approaches c, not what value (if any) f actually takes at c The next example illustrates this point rather dramatically Foundations of Calculus
Fig 23
EXAMPLE 23
De ne h(x) = Calculate limx 7 h(x) 3 1 if x = 7 if x = 7
SOLUTION It would be incorrect to simply plug the value 7 into the function h and thereby to conclude that the limit is 1 In fact when x is near to 7 but unequal to 7, we see that h takes the value 3 This statement is true no matter how close x is to 7 We conclude that limx 7 h(x) = 3 You Try It: Calculate limx 4 [x 2 x 12]/[x 4] ONESIDED LIMITS
x c
There is also a concept of onesided limit We say that lim f (x) = if the values of f become closer and closer to when x is near to c but on the left In other words, in studying limx c f (x), we only consider values of x that are less than c Likewise, we say that x c+
lim f (x) = if the values of f become closer and closer to when x is near to c but on the right In other words, in studying limx c+ f (x), we only consider values of x that are greater than c CHAPTER 2 Foundations of Calculus
EXAMPLE 24
Discuss the limits of the function f (x) = at c = 2 2x 4 x
if x < 2 if x 2
SOLUTION As x approaches 2 from the left, f (x) = 2x 4 approaches 0 As x approaches 2 from the right, f (x) = x 2 approaches 4 Thus we see that f has left limit 0 at c = 2, written x 2 lim f (x) = 0, and f has right limit 4 at c = 2, written
x 2+
lim f (x) = 4 Note that the full limit limx 2 f (x) does not exist (because the left and right limits are unequal) You Try It: Discuss onesided limits at c = 3 for the function x 3 x if x < 3 f (x) = 24 if x = 3 4x + 1 if x > 3 All the properties of limits that will be developed in this chapter, as well as the rest of the book, apply equally well to onesided limits as to twosided (or standard) limits

