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3. lim 10n + 11n3 975 + 2n 1 10/9 21/11 00 - Explain: Let N E N such that N > _ Prove lim 10n + 1In 10 On + 2n =4 00 By the Archimedean property (Theorem 3.3.9), if n > N then n 2 N > - an 9 > - and - SN = - 9 10 + 4 lim 10n + 1In On + 2n -= lim by dividing each term by n. 187 - 81 = 2 -0=25 9 9 TENS - = E (2) Claim 1: lim (10 + 14 ) = 10 Thus, lim = 0. Claim 2: lim + =0 Claim 2.1. lim ( =0 WTS lim (= 0, which means VE > 0, IN E N such that Vn Z N , - 0 = = n > / ( 2 ) VE > 0, 3N E N such that Vn 2 N , 10 + 11 - 73 - 10 Objective: Find N. i ( 2 ) Vn 2 N, 10 + 11 23 - 10 = 11 By the Archimedean property (Theorem 3.3.9), N then n 2 N > ( = ) and N = = 2 2 N5 73 +1 = n> (# ) 18n - 81=-0=2 2 2 15 S Let N E N such that N > i ( #2 ) V ( = ) By the Archimedean property (Theorem 3.3.9), Thus , lim = 0. if n > N then n 2 N > ( ) and ? = = 11 11 11 2 3 N3 Ism - $1 = 10 +4 - 10 = 11 11 11 By Theorem 4.2. 1(a), N3 S = E ( 1 ( 4 ) lim ( + 2 - lim () + lim (2) = 0+0 = 0. Thus, lim (10 + 2 ) = 10 . Thus , lim ( 2 + 2) = 0. WTS lim ( = 0, which means By Theorem 4.2.1(d), 10n + 11n3 lim (10 + 14 ) VE > 0, IN E N such that Vn Z N , - O CE . lim 10 9n5 + 2n lim ( 2+ 2) Objective: Find N. Therefore, Vn 2 N 1 9 . lim 10n + 1In3 In + 2n - = +00. . Question 2