Evaluate each of the following, using theorems on limits. in Visual Studio .NET

Creation QR Code in Visual Studio .NET Evaluate each of the following, using theorems on limits.

3.20. Evaluate each of the following, using theorems on limits.
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lim x2 6x 4 lim x2 lim 6x lim 4
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x!2 x!2 x!2
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lim x lim x lim 6 lim x lim 4
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FUNCTIONS, LIMITS, AND CONTINUITY
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lim lim x 3 2x 1 x! 1 x 3 x! 1 2x 1 2 3 3 2 4 2 x2 3x 2 lim x 3x 2
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x! 1
3 1 2 2 4 2x4 3x2 1 x x lim lim 1 3 x!1 6x4 x3 3x x!1 6 3 x x 3 1 lim 2 lim 2 lim 4 2 1 x!1 x!1 x x!1 x 1 3 6 3 lim 6 lim lim 3 x!1 x!1 x x!1 x
by Problem 3.19. p p p 4 h 2 4 h 2 4 h 2 d lim lim p h!0 h!0 h h 4 h 2 4 h 4 1 1 1 lim p lim p h!0 h 4 h 2 h!0 4 h 2 2 2 4 e p sin x sin x p sin x x lim lim x 1 0 0: lim p lim x!0 x x!0 x x!0 x
x!0
Note that in (c), (d), and (e) if we use the theorems on limits indiscriminately we obtain the so called indeterminate forms 1=1 and 0/0. To avoid such predicaments, note that in each case the form of the limit is suitably modi ed. For other methods of evaluating limits, see 4.
CONTINUITY (Assume that values at which continuity is to be demonstrated, are interior domain values unless otherwise stated.) 3.21. Prove that f x x2 is continuous at x 2.
Method 1: By Problem 3.10, lim f x f 2 4 and so f x is continuous at x 2.
Method 2: We must show that given any  > 0, we can nd  > 0 (depending on ) such that j f x f 2 j jx2 4j <  when jx 2j < . The proof patterns that are given in Problem 3.10.
3.22. (a) Prove that f x
x sin 1=x; x 6 0 is not continuous at x 0. (b) Can one rede ne f 0 5; x 0 so that f x is continuous at x 0
(a) From Problem 3.13, lim f x 0. But this limit is not equal to f 0 5, so that f x is discontinuous x!0 at x 0. (b) By rede ning f x so that f 0 0, the function becomes continuous. Because the function can be made continuous at a point simply by rede ning the function at the point, we call the point a removable discontinuity.
3.23. Is the function f x
2x4 6x3 x2 3 continuous at x 1 x 1
f 1 does not exist, so that f x is not continuous at x 1. By rede ning f x so that f 1 lim x!1 f x 8 (see Problem 3.11), it becomes continuous at x 1, i.e., x 1 is a removable discontinuity.
3.24. Prove that if f x and g x are continuous at x x0 , so also are (a) f x g x , (b) f x g x , f x if f x0 6 0. (c) g x
CHAP. 3]
FUNCTIONS, LIMITS, AND CONTINUITY
These results follow at once from the proofs given in Problem 3.19 by taking A f x0 and B g x0 and rewriting 0 < jx x0 j <  as jx x0 j < , i.e., including x x0 .
3.25. Prove that f x x is continuous at any point x x0 .
We must show that, given any  > 0, we can nd  > 0 such that j f x f x0 j jx x0 j <  when jx x0 j < . By choosing  , the result follows at once.
3.26. Prove that f x 2x3 x is continuous at any point x x0 .
Since x is continuous at any point x x0 (Problem 3.25) so also is x x x2 , x2 x x3 , 2x3 , and nally 2x3 x, using the theorem (Problem 3.24) that sums and products of continuous functions are continuous.
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