(b) Theorem: Assume that f: RR and g: RR are functions, and let c € R. Suppose that f(x) ≤ g(x) for all x ER-{c), and th

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B Theorem Assume That F Rr And G Rr Are Functions And Let C R Suppose That F X G X For All X Er C And Th 1 (22.63 KiB) Viewed 30 times

(b) Theorem: Assume that f: RR and g: RR are functions, and let c € R. Suppose that f(x) ≤ g(x) for all x ER-{c), and that lim f(x) and lim g(x) both exist. Then lim f(x) lim g(x). X-C Proof: Let {n} be any sequence such that lim n = c and an e for all n € N. (1) 318 Then lim f(x) = lim f(rn) and lim g(x) = lim g(n). (2) We have lim f(n) ≤ lim g(n). (3) 84x 84x Thus lim f(x) ≤ lim g(x). (4) PIC (i) Explain why (2) is true. (ii) Explain why (3) is true. (iii) Explain why (4) is true. [12 marks]