Question

三道高数问题，求解题及过程

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Answer 1

(1)
f(x)
=[ (1-x)/(1+x)]^(1/x) ; x≠0
=a ; x=0
lim(x->0) [ (1-x)/(1+x)]^(1/x)
=lim(x->0) e^{ ln[ (1-x)/(1+x)] /x } (0/0 分子分母分别求导)
=lim(x->0) e^[-1/(1-x) -1/(1+x)]
=e^(-1-1)
=e^(-2)
=>

a=e^(-2)
(2)
x->0
分母
ln(1+tan2x) =ln(1+2x+o(x) ) = 2x +o(x)
分子
e^x =1 +x +(1/2)x^2 +o(x^2)
(e^x -1) - x =(1/2)x^2 +o(x^2)
//
lim(x->0) ∫(0->x) [(1/t^2)(e^t -1) - 1/t] dt / ln(1+tan2x)
=lim(x->0) ∫(0->x) [(1/t^2)(e^t -1) - 1/t] dt / (2x)
(0/0 分子分母分别求导)
=lim(x->0) [(1/x^2)(e^x -1) - 1/x] / 2
=lim(x->0) [(e^x -1) - x] / (2x^2)
=lim(x->0) (1/2)x^2 / (2x^2)
=1/4

(3)
f(x) =x^2.sinx
f(-x) =-f(x)
=>
∫(-1->1) x^2.sinx dx =0

Answer 2

详情如图所示

有任何疑惑，欢迎追问