1.

lim_(Nrarroo)[(1-2sin theta)^nsum_(r=0)^n .^nC_r((sin 2 theta+sin thetasin2theta)/(2(costheta-sin2theta)))^r]^(1/theta) is equal to (where n in N)

Answer»

`e^(3n)`
`e^(-n)`
`e^(-4N)`
`e^(1//n)`

Solution :`lim_(thetararr0) [(1-2 sintheta)^nsum_(r=0)^n.^nC_r((2costheta(sintheta+SIN^2theta))/(2 COS theta(1-2 sintheta)))^r]^(1/theta)`
`lim_(thetararr0) [sum_(r=0)^n.^nC_r (sintheta+ sin^2theta)(1-2sin theta)^(n-r)]^(1/theta) `
`lim_(thetararr0) [(1-2 sintheta+sin theta+ sin^2 theta)^n]^(1/theta)`
`elim_(thetararr0) [-sin theta + sin^2theta]n/theta`
`elim_(thetararr0) (sin theta-1)n`
`e^(-n)`


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