if `Delta (x) = |{:(tan x,,tan (x+h),,tan(x+2h)),(tan(x+2h),,tan x,,tan(x+h)),(tan(x+h),,tan(x+2h),,tanx):}|, " then "`
The value of `lim_(h to 0) .(Delta (pi//3))/(sqrt(3)h^(2)) " is "`

Options

A. 144
B. 81
C. 64
D. 36

Answer

A

Explanation

`(Delta)/(h^(2)) = |{:(tan x,,(tan (x+h)-tanx)/(h),,(tan (x+2h)-tanx)/(h)),(tan(x+2h),,(tan x-tan(x+2h))/(h),,(tan(x+h)-tan(x+2h))/(h)),(tan(x+h),,(tan(x+2h)-tan(x+h))/(h),,(tanx-tan (x+h))/(h)):}|`
`rArr underset( h to 0)("lim") .(Delta)/(h^(2)) =|{:(tan x,,sec^(2) x,,2sec^(2) x),(tan x,,-2sec^(2) x,,-sec^(2) x),(tan x,,sec^(2)x,,-sec^(2) x):}|`
`= |{:(0,,0,,3sec^(2)x),(0,,-3sec^(2)x,,0),(tan x,,sec^(2)x,,-sec^(2)x):}|`
`= tan x sec^(4) x`
`rArr underset( h to 0)(" lim ") .(Delta (pi//3))/(sqrt(3)h^(2)) =144`