+tan(sinx)sin(tanx)什么意思
limx→0+tan(sinx)sin(tanx)
原式=limx→0+[sec(sinx)]^2cosx2tan(sinx)cos(tanx)sec2x2sin(tanx)(用罗比塔法则)
=limx→0+sec2(sinx)cosxcos(tanx)sec2x•limx→0+sin(tanx)tan(sinx)(分离非零极限乘积因子)
=limx→0+sin(tanx)tan(sinx)(算出非零极限)
=limx→0+cos(sinx)sec2x2sin(tanx)sec2(sinx)cosx2tan(sinx)(用罗比塔法则)
=limx→0+cos(sinx)sec2xsec2(sinx)cosx•limx→0+tan(sinx)sin(tanx)
=limx→0+tan(sinx)sin(tanx)
出现循环,此时用罗比塔法则求不出结果.怎么办?用等价无穷小代换.
∵sinx~tanx(x→0)
∴原式=limx→0+xx=1而得解.
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