Evaluate the following limit limit as x approaches 0 from the negative (cot x-(1

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ot(x) = cos(x) / sin(x) So the least common denominator of (cos(x) / sin(x)) - (1/x) is x * sin(x) and the problem is equivalent to: limit [(x * cos(x)) - sin(x)] / (x * sin(x)) x->0 If you are not familiar with l'Hopital's Rule please read the definiton in Wikipedia or your calculus textbook. Using l'Hopital's Rule we first have to differentiate the numerator and denominator of this fraction. d(((x * cos(x)) - sin(x))/dx = -x*sin(x) + cos(x) - cos(x) = -x*sin(x) d(x * sin(x)) = x*cos(x) + sin(x) Since this limit is still (0/0) we reapply l'Hopital's Rule d(-x*cos(x)) = ((x * sin(x)) - cos(x)) d(x*cos(x) + sin(x)) = ((-x * sin(x)) + cos(x) + sin(x)) limit ((-x * sin(x)) + cos(x) + sin(x)) / ((x * sin(x)) - cos(x)) = 1/(-1) = -1 x->0

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