` lim _ (x to0) (sin (pi cos^(2) x))/ (x^(2))` का मान है ।

Options

A. `(pi)/2`
B. `1`
C. `-pi`
D. `pi`

Answer

D

Explanation

` lim _( x to 0) (sin ( pi cos^(2) x))/(x^(2))`
` = lim _( x to 0 ) (sin pi (1 - sin^(2) x ))/ (x^(2))`
` = lim _( x to 0 ) (sin ( pi - pi sin^(2) x ))/(x^(2))`
` = lim _( x to 0 ) ( sin ( pi sin^(2) x ))/(x^(2)) " " [ :' sin ( pi - x ) = sin x] `
` lim _( x to 0 ) ( sin pi ( sin^(2) x))/ (x^(2)) `
` lim _( x to 0 ) ( sin pi ( sin^(2) x))/(pi sin^(2) x ) xx pi ((sin^(2) x)/x^(2))" " [ :' lim _( x to 0 ) (sin theta )/theta = 1 ] `
` = 1 xx pi ( 1) `
` = pi `
अतः विकल्प (iv ) `pi` सही है ।