The value of `lim_(x to oo)((pi)/(2)-tan^(-1)x)^(x//1)` is

Options

A. 1
B. 0
C. -1
D. e

Answer

B

Explanation

` y = lim _ ( x to oo) ( (pi )/( 2 ) - tan ^( -1 ) x )^( 1//x ) `
` rArr log y = lim _ ( x to oo) ( 1)/( x) log (( pi )/( 2 ) - tan ^( - 1 ) x ) " " [ ( oo) /( oo) ` form]
` = lim _ ( x to oo) (( - ( 1 )/( 1 +x ^ 2 )))/ ( pi ) /( 2) - tan ^( -1 ) x ) `
[ using L ' Hosipital's rule ]
` = lim _ ( x to oo) ( ( 2x)/( ( 1 + x ^ 2 )^ 2 ) /( - ( ( 1 )/( 1 + x ^ 2 ))) `
` = lim _ ( x to oo) ( - 2x ) /( 1 + x ^ 2 ) ` [ using L' Hospital's rule ]
` rArr log y = 0 rArr y = 1 `