lim x→0 x(e^2x-1)/(tgx)^2 lim x→0 (tanx-sinx）/(sin3x)^2 lim x→0 ln(1-3x^2)/2xsin3x

题目

lim x→0 x(e^2x-1)/(tgx)^2 lim x→0 (tanx-sinx）/(sin3x)^2 lim x→0 ln(1-3x^2)/2xsin3x

答案

lim (x(e^(2x)-1))/(tgx)^2 = lim((e^(2x)-1)/x)/(tgx/x)^2=2/(1)^2=2
lim x→0 ln(1-3x^2)/2xsin3x=-6x/(1-3x^2)/(2sin3x+6xcos3x)=-6/(1-3x^2)[1/(6cos3x+6cos3x-18xsin3x)]=-6/12=-1/2
lim x→0 (tanx-sinx）/(sin3x)^2=(sec^2 x-cosx)/3(sin6x)=[2 secx tgx secx+sinx]/18cos6x=0