InterviewSolution
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InterviewSolution
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`lim_(x->pi/2` `(cot x - cos x)/(pi-2x)^3` equals |
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Answer» Let `x = pi/2 -h`, then the given expression becomes, `lim_(h->0) (cot(pi/2-h) - cos(pi/2-h))/((pi- 2(pi/2-h))^3)` `=lim_(h->0) (tanh - sinh)/((2h)^3)` `=lim_(h->0) ((sinh/cosh) - sinh)/(8h^3)` `=lim_(h->0) (sinh((1-cosh)/cosh))/(8h^3)` `=lim_(h->0) (tanh(2sin^2(h/2)))/(8h^3)` `=lim_(h->0) tanh/h*(sin^2(h/2))/(4(h/2)^2*4)` `= 1*1^2/(4*4) = 1/16`, which is the required value. |
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