①limx→0(x+e^3x)^1/x

题目

①limx→0(x+e^3x)^1/x
②limx→0(e^x－e^sinx)/(sinx)^3
③limx→0(e^-1/x^2)/x^100

答案

①limx→0(x+e^3x)^1/x
=lim[e^ln(x+e^3x)^1/x
=e^lim[ln(x+e^3x)/x]
=e^lim[(1+3e^3x)/(x+e^3x)] 罗比达
=e^4
②limx→0(e^x－e^sinx)/(sinx)^3
=lime^x[1－e^(sinx-x)]/(sinx)^3
=lime^x*lim[sinx-x]/x^3 等价替换+四则运算
=lim[cosx-1]/3x^2 罗比达
=lim-sinx/6x 罗比达
=-1/6
③limx→0(e^-1/x^2)/x^100
=lim(e^-1/x^2)/x^100 令1/x^2=t
=lim(t→+∞)(e^-t)*t^50
= lim t^50/e^t
=0 用50次罗比达法则