Friday, 20 February 2015

calculus - Why is limlimitsxto0+xcotx=1?



Why is limx0+xcotx=1?



Since both x and cotx 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: limx0+xcotx=00=0?



PS: I know how to find the limit: limx0xcotx=limx0xcosxsinx=limx0cosx=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/tanx. The tangent is 0 at 0 so its reciprocal has a pole at 0.



It is important to note that while the cotangent is (tanx)1 this is not the same as tan1(x), the inverse function (the functional inverse) of the tangent also called arcus tangent. This would be 0 at 0.


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