Given:
limx→09x−2⋅6x+4xx2
=limx→0(3x)2−2⋅(3x×2x)+(2x)2x2
=limx→0(3x−2x)2x2
[∵(a−b)2=a2−2ab+b2]
=[limx→03x−2x+1−1x]2
=[limx→0(3x−1)−(2x−1)x]2
=[limx→03x−1x−limx→0(2x−1)x]2
=[limx→03x−1x−limx→0(2x−1)x]2
∵limx→0[(ax−1x)=loga]
=[log3−log2]2
=[log32]2
Therefore,
limx→09x−2.6x+4xx2=[log32]2