By M. H. Protter C. B. Morrey Jr.
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Extra info for A First Course in Real Analysis
Suppose that 1; (x) ->; Li as x ->; a for i = 1, 2, ... , n, and suppose that g(x) = fl(x) ... fn(x). Then limHag(x) = LIL2 ... 7 (Limit of a composite function). Suppose that f and g are functions on ~I to ~I' Define the composite function h(x) = f[g(x)]. Iff is continuous at L, and ifg(x)->;L as x->;a, then lim h(x) = f(L). x->a PROOF. Since f is continuous at L, we know that for every f This proof assumed that L2 L 2 =O. 2 > 0 there is a '* O. A slight modification of the proof establishes the result if 39 2: Continuity and limits 8 1 > 0 such that < 10 If(t) - f(L)1 whenever It - LI < 8 1, From the fact that g(x) ~ L as x ~ a, it follows that for every is a 8 > 0 such that Ig(x) - LI < 10' whenever 0 < Ix - al < 8.
A set S of numbers is said to be inductive if and only if (a) 1 E Sand (b) (x + 1) E S whenever xES. Examples of inductive sets are easily found. The set of all real numbers is inductive, as is the set of all rationals. The set of all integers, positive, zero, and negative is inductive. The collection of real numbers between 0 and 10 is not inductive since it satisfies (a) but not (b). No finite set of real numbers can be inductive since (b) will be violated at some stage. Definition. A real number n is said to be a natural number if and only if it belongs to every inductive set of real numbers.
Find the values of p, if any, for which the following limits exist. (a) limx--+o+x P sin(l / x) (b) limx .... +"'xPsin(l/x) (c) limx--+_ ",lxlP sin(l / x) 19. 9 for the case when x ~ + 00 (instead of x~a). 20. 10 (Sandwiching theorem) for the case when x ~ - 00 (instead of x ~ a). 21. 18.
A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.