Question

`lim_(x-pi/2) (cot x - cosx)/(pi-2x)` equals: (1) `1/8` (2) `1/4` (3) `1/24` (4) `1/16`

Answer 1

Let `x = pi/2 - h`,
Then, our limit becomes,
`Lim_(h->0) (cot(pi/2-h) - cos(pi/2-h))/(pi -(2(pi/2-h))^3 )`
`=Lim_(h->0) (tanh - sinh)/(pi -2(pi/2-h))^3 `
`=Lim_(h->0) (sinh(1-cosh)/(cosh))/(8h^3) `
`=Lim_(h->0) (tanh(2sin^2(h/2)))/(8h^3) `
`=Lim_(h->0) (tanh(sin^2(h/2)))/(4h^3) `
`=Lim_(h->0) (tanh/h)(sin^2(h/2))/(4*4(h/2)^2) `
`=1*1/16*1 = 1/16`
`:. ` Option `(4)` is the correct option.