Question

lim(x趋向于无穷)(2x-3/2x 1)^x-1

Answer 1

lim (1/x2 - cot2x)，x趋向0 = lim (1/x2 - cos2x/sin2x)，通分 = lim (sin2x - x2cos2x)/(x2sin2x)，分母sin2x等价x2 = lim cos2x(tan2x - x2)/x^4，分子提取cos2x出来并注意lim cos2x = 1 = lim (tanx - x)(tanx + x)/x^4，平方差 = lim (tanx - x)/x3 * lim (tanx + x)/x，第一项用洛必达法则上下求导 = lim (sec2x - 1)/3x2 * ( lim tanx/x + 1)，tanx等价x = lim tan2x/3x2 * (1 + 1)，恒等式sec2x - 1 = tan2x = lim x2/3x2 * 2，tan2x等价x2 = 2/3

Answer 2

如图

Answer 3

如下