Question

计算下列极限（要过程）

Answer 1

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