` lim _ ( x to 0) ( x cos x - sin x )/( x^(2) sin x) `का मान ज्ञात कीजिए ।

Options

A. `1/2`
B. `-1/2`
C. `1/3`
D. `-1/3`

Answer

D

Explanation

` lim _( x to 0) ( x cos x - sin x) /( x^(2) sin x) `
` = lim_ ( x to 0) ( x ( 1- x^(2)/(2i) + x^(4)/(4i) - . . . . . )- ( x - x^(3)/(3!) + x^(5)/(5 !)- . . . . . ))/(x^(2) ( x - x^(3) /(3!) + x^(5)/(5!) - . . . . . ))`
` lim _( x to 0) ( x - x^(3) /(2!) + x^(5)/(4!) - . . . . - x+ x^(3)/( 3!) - x^(5)/( 5!)+ . . . . . )/( x^(3) ( 1- x^(2)/(3!) + x^(4) /(4!) . . . . .))`
` = lim _ ( x to 0) ( x^(3) (-1/(2!) + 1/(3!))+ x^(5) ( 1/(4!)- 1/(5 !)) + . . . . )/( x^(3) ( 1- x^(2)/(3!)+ x^(4)/(5!)-. . . . ))`
` = lim _ ( x to 0) ((-1/(2 !)+1/(3!))+ x^(2) ( 1/(4!) - 1/(5!))+ . . . . . .)/( 1-x^(2)/(3!)+ x^(4)/(5!) - . . . . . )`
` = ( -1/(2!)+1/(3!))/1 = -1/2 + 1/6 = (-3+ 1) /6 = -2/6 = -1/3.`