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).

Calculate the following limits of functions using the sequential character- ization from Theorem 4.2.3 (and theorems about limits of sequences, if required). (a) lim +1. x 4 2+31 (b) lim ²-1 2-1 2-1' (c) lim 4x³ 3x² + 7x - 11. x-0

4" Question 2 Use the Squeeze Theorem to find the limit of the sequence n!

Show that the converse of Theorem Limit of a Sequence is not true.

Prove that limit x  3   (4x-5) =7

Question 6 Answer the following. Do the best you can entering in mathematical statements into the answer field. 1. State the Squeeze Theorem for sequences. 2. Explain how the connection between functional limits and sequential convergence allows one to deduce the Squeeze Theorem for limits from the Squeeze Theorem for sequences. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). BIUS Paragraph Open Sans,sa... V 10pt V Ev A AV Ix XQ

please answer both of these calculus grade 12 questions. The first question with two parts is very short. That's why Im asking if you can also answer the next question too.

Find the limit

8) find the limit

In one plane, draw the closed second quadrant Q2 along with the first three terms of the sequences defined by an = Inn and b,n = 1 + i(rn/4). Next, let f(2) = e² and in another plane draw f(Q2) along with the first three terms of f(an) and f(b,). Does either lim,+0 f(an) or lim,→ ƒ(bn) exist?