Evaluate limit X tends to zero bracket 1 upon x minus cot X

Question

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Answer 1

lim_x→0 [1/x - cotx] = lim_x→0 [1/x - cosx/sinx]

 

       = lim_x→0 [(sinx - xcosx) / (xsinx)] (0/0 form)

 

       = lim_x→0 [(cosx - cosx +xsinx) / (sinx + xcosx)] (By L'Hopital's Rule)

 

       = lim_x→0 [xsinx / (sinx + xcosx)] (Still has the form 0/0)

 

       = lim_x→0 [(sinx + xcosx) / (cosx + cosx - xsinx)] (By L'Hopital's Rule)

 

       = 0 / 2 = 0