lim(x-0) (sinx/x)^ctg^2(x)

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/24 03:23:57
lim(x-0)(sinx/x)^ctg^2(x)lim(x-0)(sinx/x)^ctg^2(x)lim(x-0)(sinx/x)^ctg^2(x)(sinx/x)^ctg^2(x)=e^[ln(s

lim(x-0) (sinx/x)^ctg^2(x)
lim(x-0) (sinx/x)^ctg^2(x)

lim(x-0) (sinx/x)^ctg^2(x)
(sinx/x)^ctg^2(x)
=e^[ln(sinx/x)^ctg^2(x)]
=e^[ctg^2(x)*ln(sinx/x)]
=e^[ln(sinx/x)/tg^2(x)]
lim(x->0) ln(sinx/x)/tg^2(x) 分子分母等价无穷小替换
=lim(x->0) (sinx/x-1)/x^2
=lim(x->0) (sinx-x)/x^3 分子分母同趋向于0,运用洛必达法则
=lim(x->0) (cosx-1)/3x^2
=lim(x->0) -sinx/6x
=-1/6
所以原式=e^{lim(x->0) ln(sinx/x)/tg^2(x)}
=e^(-1/6)