Question

急求解4道关于极限的题目

1.lim(1/(x+1)-3/(x^3+1)) (x趋向于-1)
2.lim((1-cos3x)/x^2) (x趋向于0)
3lim(((1+3x)/3x)^-2) (x趋向于无穷大)
4.lim((1-3x)^(1/x)) (x趋向于0)

要有解题过程

Answer 1

如图，图片需要审核，稍安勿躁

Answer 2

第一题通分，第二题用等价无穷小替换，第三四题用重要极限。过程太难输入，你再问问身边的同学老师吧

Answer 3

-3
2
-1
e^3

Answer 4

lim[1/(x+1)-3/(x^3+1)] (x趋向于-1)
=lim[(x^2-x+1)/(x^3+1)-3/(x^3+1) ](x趋向于-1)
=lim[(x^2-x-2)/(x^3+1)](x趋向于-1)
=lim{(x-2)(x+1)/[(x+1)(x^2-x+1)]}(x趋向于-1)
=lim(x-2)/(x^2-x+1)(x趋向于-1)
=(-1-2)/(1+1+1)
=-1

lim[(1-cos3x)/x^2] (x趋向于0)
=lim2[sin(3x/2)]^2/x^2] (x趋向于0)
=lim2(3x/2)]^2/x^2] (x趋向于0)
=9/2

lim[(1+3x)/3x]^(-2) (x趋向于无穷大)
=lim[(1+1/(3x)]^(-2) (x趋向于无穷大)
=[(1+0]^(-2)
=1

lim[(1-3x)^(1/x)] (x趋向于0)
=lim{(1-3x)^[1/(-3x)]}^(-3) (x趋向于0)
=e^(-3)