Evaluatelimx→0sin2x(2-1+cosx):
2
4
42
Explanation for correct option
Evaluating the given limit expression
Given, the expression is: limx→0sin2x(2-1+cosx)
Solving we get,
=limx→0sin2x(2-1+cosx),=limx→0sin2x(2-1+cosx)×(2+1+cosx)(2+1+cosx)=limx→0(sin2x)1×(2+1+cosx)((2)2-(1+cosx)2)[∵(a+b)(a-b)=a2-b2]=limx→0(sin2x)×(2+1+cosx)(2-(1+cosx))=limx→0(sin2x)×(2+1+cosx)(1-cosx)=limx→0(sin2x)1×x2x2×(2+1+cosx)(1-cosx)=limx→0(sin2x)x2×x21×12sin2(x2)×(2+1+cosx)[∵(1-cosx)=(2sin2(x2))]=limx→0(sin2x)x2×x21×12sin2(x2)×(2+1+cosx)=limx→0(sin2x)x2×(x24)×42sin2(x2)×(2+1+cosx)=limx→0(sin2x)x2×limx→0(1sin2(x2)(x24))×42×limx→0(2+1+cosx)L=limx→0(sinxx)2×limx→0(1(sin(x2)(x2))2)×2×limx→0(2+1+cosx)L=1×1×2×22[∵limt→0sintt=1]L=42
Hence, the correct answer is option (D).
Evaluate the determinants.
∣∣ ∣∣012−10−3−230∣∣ ∣∣