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lim(x->pi/2 (cot x - cos x)/(pi-2x)^3 eq...

`lim_(x->pi/2` `(cot x - cos x)/(pi-2x)^3` equals

Text Solution

Verified by Experts

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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