मान ज्ञात कीजिए -
` lim_( x to pi //4)(sin x - cos x)/(x - pi/4)`

Options

A. `0`
B. `1`
C. `sqrt(2) `
D. `1/sqrt(2) `

Answer

C

Explanation

` lim_( x to pi //4 ) ( sin x - cos x)/( x- pi/4 )`
` x = pi/4 + h ` रखने पर , जब ` x to pi/4` तब `h to 0`
`= lim _ ( h to 0) ( sin ( pi /4 + 4) - cos ( pi/4 + h))/( (pi/4 + h ) - pi/4)`
` = lim _( h to 0) (( sin. pi/4 cos h + cos . pi/4 sin h ) - ( cos. pi/4 cos h - sin. pi/4 sin h))/h`
` = lim _( h to 0) ((( cos h)/sqrt(2) + (sinh)/sqrt(2) ) - ( (cosh)/sqrt(2)-(sin h)/sqrt(2)))/h`
` = lim _ ( h to 0) (( 2 sin h)/(sqrt(2)))/h`
` = lim _( h to 0) sqrt(2) (sin h)/h `
` = sqrt(2) xx [ lim _( h to 0) ( sin h)/h] " " [ :' lim _( h to 0) ( sin h ) /h = 1 ] `
` = sqrt( 2) xx 1 = sqrt(2) `