Question

计算下列极限 要过程

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

Answer 1

1.(x+e^3x)^1/x=e^[ln(1+x+e^3x-1)/x
而limln(1+x+e^3x-1)/x=lim(x+e^3x-1)/x=lim(1+3e^3x)=4
故原式=e^4
2.原式=lime^sinx[e^(x-sinx)-1]/(sinx)^3=lim(x-sinx)/x^3=lim(1-cosx)/3x^2=lim(1/2x^2)/3x^2=1/6
3.设t=1/x趋近于无穷
则原式=limt^100/e^(t^2)
多次运用罗比达法则的最后极限为0

Answer 2

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