Could someone please check my work

Prove limn in2 - An 8+ lim 87 + 5n 7n2 - 4n = lim 7- 4 - by dividing each term by me Claim 1: lim 8 + =8 Claim 2: lim 7 - = 7. WTS lim (8 + - - 8) =8, which by the definition of convergence means, VE > 0, 3N E N such that Vn 2 N . 8 + 2 - 8 on | en n Let N E N such that N > 5 By the Archimedean property (Theorem 3.3.9), if n > N then n 2 N > - and n 4 Let N E N such that N > By the Archimedean property (Theorem 3.3.9), A if n > N then n 2 N > - and - S N S N 187 - 81= 7 -2-7=4 4 4 NS = E (4) Thus, lim ( 7 - ) - 7 By Theorem 4.2.1(d), 8n2+ 5n lim (8+ ) lim 7n2 - 4n which is true because lim (7 - lim (7 - ) # 0 and 7 # 0. Therefore, 8n + 5n lim 772 - An