Why is limx→0+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: limx→0+xcotx=0⋅0=0?
PS: I know how to find the limit: limx→0xcotx=limx→0xcosxsinx=limx→0cosx=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 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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