If `f (x) =|{:(sin x , cos x, tan x),(x^(3), x^(2),x),(2x,1,1):}|, then lim_(x to 0) ""(f(x))/(x^(2))` is

Options

A. 3
B. `-1`
C. 0
D. 1

Answer

D

Explanation

Given that,
` f ( x ) = |{:( sinx,, cos x,, tan x ) ,( x^ 3,, x^ 2,, x ) ,( 2x,, 1 ,,1 ) :}| `
` rArr f ( x ) = ( x^ 2 - x ) sin x - ( x ^ 3 - 2x^ 2 ) cos x - x ^ 3 tan x `
` = x ^ 2 sin x - x ^ 3 cos x - x ^ 3 tan x + 2x^ 2 cos x - x sin x `
` therefore lim _ ( x to 0 ) ( f (x ) ) /( x ^ 2 ) = lim _ (x to 0 ) `
` ( sin x - x cos x - x tan x + 2 cos x - (sin x ) /( x ) ) `
` = 0 - 0 -0 + 2 - 1 = 1 `