Question

`lim_(x rightarrow π/4) (8sqrt(2) - (cosx + sinx)^7)/(sqrt(2) - sqrt(2)sin2x)` is equal to ______.

विकल्प

• 14

• 7

• `14sqrt(2)`

• `7sqrt(2)`

Answer 1

`lim_(x rightarrow π/4) (8sqrt(2) - (cosx + sinx)^7)/(sqrt(2) - sqrt(2)sin2x)` is equal to 14.

Explanation:

`lim_(x rightarrow π/4) (8sqrt(2) - (cosx + sinx)^7)/(sqrt(2) - sqrt(2)sin2x)` ...`["from" 0/0]`

∴ Apply L-hospital rule

`lim_(x rightarrow π/4) (-7(cosx + sinx)^6(cosx - sinx))/(-2sqrt(2)cos2x)` ...`["from" 0/0]`

`lim_(x rightarrow π/4) (7(cosx + sinx)^6(cosx - sinx))/(2sqrt(2)(cosx - sinx)(cosx + sinx))`   ...[∵ cos²x = cos²x – sin²x]

`lim_(x rightarrow π/4) (7(cosx + sinx)^6)/(2sqrt(2)(cosx + sinx))`  

= `(7(sqrt(2))^6)/(2sqrt(2)(sqrt(2))` 

= `(7 xx 8)/4`

= 14