Exercise 1: Suppose lim f(x)= xC = L, lim g(x) xC - M and there exists...

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Exercise 1: Suppose lim f(x)= x→C = L, lim g(x) x→C - M and there exists a positive number 8 such that f(x) g(x) Vx € (c-d,c + 8)\{c}. Prove that L > M. Give an example such that f(x) > g(x) \x ≤ R but lim f(x) and lim g(x) both exist and are equal. X-C x-C Exercise 2: Suppose lim f(x) = 0 and there exists positive numbers p and M such that x-C g(x) ≤M Vx € (c-p,c + p)\{c}. _sin ¹ Prove that lim f(x)g(x) = 0. Use this result to show that lim tan(x) es x→C = 0 Exercise 1: Suppose lim f(x)= x→C = L, lim g(x) x→C - M and there exists a positive number 8 such that f(x) g(x) Vx € (c-d,c + 8)\{c}. Prove that L > M. Give an example such that f(x) > g(x) \x ≤ R but lim f(x) and lim g(x) both exist and are equal. X-C x-C Exercise 2: Suppose lim f(x) = 0 and there exists positive numbers p and M such that x-C g(x) ≤M Vx € (c-p,c + p)\{c}. _sin ¹ Prove that lim f(x)g(x) = 0. Use this result to show that lim tan(x) es x→C = 0

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ANSWER Exercise 1 We can use proof by contradiction to prove that L M So assume that L M Then we have L M 0 Since both L and M exist we can use the li... View the full answer