Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits). Let f and g be functions defined on a domain ACR, and assume lime f(x) = L and lime 9(x)=M for some limit point c of A. Then, (i) lim kf(x) LC = kL for all kЄ R, (ii) lim [f(x)+9(a)] = L+M, C (iii) lim [f(x)g(x)] = LM, and 91F (iv) lim f(x)/9(x)=L/M, provided M +0. Theorem 4.2.3 (Sequential Criterion for Functional Limits). Given a function f: AR and a limit point c of A, the following two statements are equivalent: (i) lim f(x) = L. 01-2 (ii) For all sequences (an) C A satisfying xn c and (xn)c, it follows that f(xn) → L. 23 The Algebraic and Order Limit Theorems (i) To prove this statement, we need to argue that the quantity (an+bn)-(a+b)| 51 can be made less than an arbitrary e using the assumptions that (an-al and - can be made as small as we like for large n. The first step is to use the triangle inequality (Example 1.2.5) to say (an+bn)-(a+b)| = |(ana) + (bnb)|≤lan-al + lb-bl. Again, we let >0 be arbitrary. The technique this time is to divide the e between the two expressions on the right-hand side in the preceding inequality. Using the hypothesis that (an) a, we know there exists an N₁ such that lan-al< whenever n≥ N₁. Likewise, the assumption that (bn) b means that we can choose an N₂ so |bb|< whenever n≥ N₂. that - The question now arises as to which of N₁ or N₂ we should take to be our choice of N. By choosing N = max(N1, N2), we ensure that if n ≥ N, then > N₁ and n ≥ №2. This allows us to conclude that (an+bn)-(a+b)| ≤an-a|+|bn-b € for all n≥N, as desired. < + = € 2 Exercise 4.2.1. (a) Supply the details for how Corollary 4.2.4 part (ii) follows from the Sequential Criterion for Functional Limits in Theorem 4.2.3 and the Algebraic Limit Theorem for sequences proved in Chapter 2. (b) Now, write another proof of Corollary 4.2.4 part (ii) directly from Defini- tion 4.2.1 without using the sequential criterion in Theorem 4.2.3. (c) Repeat (a) and (b) for Corollary 4.2.4 part (iii).

Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits). Let f and g be functions defined on a domain ACR, and assume lime f(x) = L and lime 9(x)=M for some limit point c of A. Then, (i) lim kf(x) LC = kL for all kЄ R, (ii) lim [f(x)+9(a)] = L+M, C (iii) lim [f(x)g(x)] = LM, and 91F (iv) lim f(x)/9(x)=L/M, provided M +0. Theorem 4.2.3 (Sequential Criterion for Functional Limits). Given a function f: AR and a limit point c of A, the following two statements are equivalent: (i) lim f(x) = L. 01-2 (ii) For all sequences (an) C A satisfying xn c and (xn)c, it follows that f(xn) → L. 23 The Algebraic and Order Limit Theorems (i) To prove this statement, we need to argue that the quantity (an+bn)-(a+b)| 51 can be made less than an arbitrary e using the assumptions that (an-al and - can be made as small as we like for large n. The first step is to use the triangle inequality (Example 1.2.5) to say (an+bn)-(a+b)| = |(ana) + (bnb)|≤lan-al + lb-bl. Again, we let >0 be arbitrary. The technique this time is to divide the e between the two expressions on the right-hand side in the preceding inequality. Using the hypothesis that (an) a, we know there exists an N₁ such that lan-al< whenever n≥ N₁. Likewise, the assumption that (bn) b means that we can choose an N₂ so |bb|< whenever n≥ N₂. that - The question now arises as to which of N₁ or N₂ we should take to be our choice of N. By choosing N = max(N1, N2), we ensure that if n ≥ N, then > N₁ and n ≥ №2. This allows us to conclude that (an+bn)-(a+b)| ≤an-a|+|bn-b € for all n≥N, as desired. < + = € 2 Exercise 4.2.1. (a) Supply the details for how Corollary 4.2.4 part (ii) follows from the Sequential Criterion for Functional Limits in Theorem 4.2.3 and the Algebraic Limit Theorem for sequences proved in Chapter 2. (b) Now, write another proof of Corollary 4.2.4 part (ii) directly from Defini- tion 4.2.1 without using the sequential criterion in Theorem 4.2.3. (c) Repeat (a) and (b) for Corollary 4.2.4 part (iii).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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