(b) Theorem Let f: RR and g: RR be functions and a € R. Assume that lim f(x)= L and lim g(x)= M. Then lim (f+g)(x) = L +

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B Theorem Let F Rr And G Rr Be Functions And A R Assume That Lim F X L And Lim G X M Then Lim F G X L 1 (70 KiB) Viewed 10 times

(b) Theorem Let f: RR and g: RR be functions and a € R. Assume that lim f(x)= L and lim g(x)= M. Then lim (f+g)(x) = L + M. x x→a HIG (Recall that (f+g)(x) = f(x) + g(x).) Proof: Let {n} be a sequence in R such that lim n = a and an a for all n E N. 72-00 (1) (2) (3) (4) (5) Then lim f(n) = L and lim g(n) = M. 11-00 72-00 Also lim f(x) + g(n) = lim f(n) + lim_g(n). 72-00 71-00 So lim f(n) + g(x)=L+M. 11-00 Thus lim (f+g)(x) = L + M. 71-00 And so lim (f+g)(x) = L + M. x→a (i) Explain why (1) is true. (ii) Explain why (2) is true. (iii) Explain why (4) is true. (iv) Explain why (5) is true. [16 marks]