The value of limx→0 2x3(tan⁡x−sin⁡x)2x2 is

limx→0 1+2x3(tan⁡x−sin⁡x)−12/x2=elimx→02(tan⁡x−sin⁡x)−x3⋅2x5 =elimx→022x−x33+2x515−⋯−x−x33!+x55!⋯−x3x5=elimx→0 22x58+x7 and higher powers of xx5=e1/2=e