Use Definition 2.4.1 to prove each of the limit statements in Exercise 1. 1. In each of the following, a limit statement is given. In each case, answer the following questions: (i) After how many terms are we guaranteed that an > 100 (or xn < -100)? (ii) For arbitrary but unknown M > 0, after how many terms are we guaranteed that an > M (or xn < −M)? n² +1 (a) lim √n +[infinity] = (b) lim = +[infinity] 818 n→[infinity] n + 1 (c) lim 1-n √n (d) lim n-x 818 = -[infinity] 1+4n-n³ 3n || 1 -[infinity] Definition 2.4.1 Suppose {n} is a sequence of real numbers. We say that (a) {n} diverges to +[infinity] (lim n = +[infinity]) if n→[infinity] VM > 0, 3 no EN3nZno⇒ xn > M; (b) {n} diverges to - (lin lim xn = MEN⇒ n ≥no ⇒ xn <-M. n-x Note that this definition implies that if lim xn = +[infinity] (or -[infinity]) then {n} is unbounded, hence {n} cannot converge (why?). So we will not say that {n} converges to +[infinity] or -[infinity], or that lim xn exists in these cases, even though we n-x use the notation lim xn. n48