lim x-2/3 log(1-x)/2x-3x^2 and lim approaches -pie from the left x csc x Solutio

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good question: This should be lim(x--> infinity) (x+2/x)^(3x). If you don't know the limit for e, you can still derive this limit as follows. Let L = lim(x--> infinity) (x+2/x)^(3x). Taking ln's yields ln L = lim(x--> infinity) 3x * ln(x + 2/x) = lim(x--> infinity) 3 ln(x + 2/x) / (1/x). Applying L'Hopital's Rule yields ln L = lim(x--> infinity) [3/(x + 2/x) * (1 - 2/x^2)] / (-1/x^2) = lim(x--> infinity) [-3x^2/(x + 2/x) * (1 - 2/x^2)] = lim(x--> infinity) [-3/(1 + 2/x^3) * (1 - 2/x^2)] = -3 * 1 = -3. Thus, L = e^(-3).

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