A First Course in Real Analysis (2nd Edition) (Undergraduate by Murray H. Protter, Charles B. Morrey Jr.

By Murray H. Protter, Charles B. Morrey Jr.

Many alterations were made during this moment version of A First direction in genuine Analysis. the main obvious is the addition of many difficulties and the inclusion of solutions to many of the odd-numbered workouts. The book's clarity has additionally been better by way of the additional explanation of a number of the proofs, extra explanatory comments, and clearer notation.

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For limits at infinity, a theorem on obvious limits takes the following form. 14 (Obvious limit). If f(x) = l/x for all x # 0, then lim f(x) = 0, x-s co lim f(x) = 0, x-e +00 lim f(x) = O. 7 on composite functions. 15 (Limit of a composite function). R 1. Iff is continuous at Land g(x) -+ L as x -+ +00, then lim f[g(x)] = f(L) . x-+oo Remarks. 15 may also be stated with x -+ -00 or x -+ 00 . 2 has three analogs according as x -+ 00, x -+ +00, or x -+ -00 . 7, respectively. 5 and the function f : x -+ xj(x + 1), x # -1, we see that f increases without bound as x tends to - 1from the left.

Any nonempty set T of natural numbers contains a smallest element. PROOF. Let k be a member of T. We define a set S of natural numbers by the relation S = {p: pET and p ~ k}. The set S contains a portion (not necessarily all) of the set consisting of the natural numbers {1, 2, 3, ... , k - 1, k}. Thus since S is finite it has a smallest element, which we denote by s. We now show that s is the smallest element of T. First, since s E Sand SeT, then SET. Suppose t is any element of T different from s.

F(x) = 3x - 2 { 2x - 1, 8. f(x) = {C ~ 2 1 < x < 2, 2 ~ x < 4, 2 ;:: : 2r 1 < x < 2, , x 2 /2, 9. f(x) = { to. f(x) = { ( : : : x:f. 2, :r . ), x ~3 2 < x < 3, x:f. 0, X 12. f(x) = 3" x < 4, a sm - , x { a=2 x=2, x - 9, 11. f(x) = a= 2 2 ~ x < 5, x: - 8, x - 4 4, a=2 x:f. 0, a=O a=O x=O, 13. Given f(x) = tan x, -rc/2 < x < rc/2, g(x) = (rc/2) - x. What conclusion can we draw about lim f[g(x)] when a = rc/4? Same problem when a = rc/2, when a = O. 14. 4 on the limit of equal functions for limits from the left.

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