lim(1+x^2-cos2x)^(1/lnx),x趋近于0+
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/05/10 23:39:53
lim(1+x^2-cos2x)^(1/lnx),x趋近于0+
lim (1 + x^2-cos2x) ^ (1/lnx) (x—>0+)
=lim [ x^2 + 2(sinx)^2 ] ^ (1/lnx)
=lim [ x^2(1 + 2(sinx/x)^2) ] ^ (1/lnx)
=lim (x^2)^(1/lnx) * lim [(1 + 2(sinx/x)^2) ]^(1/lnx)
=lim (x^2)^(1/lnx) * 3^0 (x—>0+)
=lim (e ^ 2t) ^ (1/t) (t=lnx,t—>-∞)
=e^2
=lim [ x^2 + 2(sinx)^2 ] ^ (1/lnx)
=lim [ x^2(1 + 2(sinx/x)^2) ] ^ (1/lnx)
=lim (x^2)^(1/lnx) * lim [(1 + 2(sinx/x)^2) ]^(1/lnx)
=lim (x^2)^(1/lnx) * 3^0 (x—>0+)
=lim (e ^ 2t) ^ (1/t) (t=lnx,t—>-∞)
=e^2
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