Friday, 20 February 2015

calculus - Why is limlimitsxto0+xcotx=1?



Why is lim?



Since both x and \cot x are continuous at zero and both equal to zero at x=0 why is the limit of both of them 1?



i.e why isn't it: \lim\limits_{x\to0+}x\cot x=0\cdot 0 = 0?



PS: I know how to find the limit: \displaystyle\lim_{x\to0}x\cot x=\lim_{x\to0}\frac {x\cos x} {\sin x}=\lim_{x\to0} \cos x = 1 and it's the same with LHR too but I just find it strange since both of them are supposed to be 0.


Answer




The cotangent is the reciprocal (the multiplicative inverse) of the tangent, that is 1/ \tan x. The tangent is 0 at 0 so its reciprocal has a pole at 0.



It is important to note that while the cotangent is (\tan x)^{-1} this is not the same as \tan^{-1} (x), the inverse function (the functional inverse) of the tangent also called arcus tangent. This would be 0 at 0.


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