1.lim x→0,(tanx-sinx)/(sin2x)^3 2.lim x→0,sin(x-1)tan(x-1)/2

1.分子分母同时乘以1/n^3
lim n→∞ (n2+5)/(n+1)(n+2)(n+3)
=lim n→∞ (1/n+5/n^3)/(1+1/n)(1+2/n)(1+3/n)
n→∞时,1/n,5/n^3均趋向0.
分子中,1/n,2/n,3/n趋向0
所以原式=0
lim x→0 ((x+sinx)/x
lim x→0 x/x+ lim x→0 sinx/x
=1+1
=2
lim x→0 ((1-cosx)/x^2
=lim x→0 2[sin(x/2)]^2/x^2
=lim x→0 2(x/2)^2/x^2
= 2*(1/2)^2
=1/2
4.y'=(1/x^2)'=(x^(-2))'=-2x^(-3)
5.
e^y-e^x+xy=0
隐函数求导,
对x求导,
d(e^y)/dx-d(e^x)/dx+d(xy)/dx=0
化为：
e^y*dy/dx-e^x+y+xdy/dx=0
则,
dy/dx=(e^x-y)/(e^y+x)

1）lim(x→∞) sinx/x+1=0+1=1 .
lim(x→∞) sinx/(x+1)=0 .
2）由于 (1+2/x)^x+3={[1+1/(x/2)]^(x/2)}^2+3 ,所以极限为 e^2+3 .
由于 (1+2/x)^(x+3)={[1+1/(x/2)]^(x/2)}^2*(1+2/x)^3 ,所以极限为 e^2*1=e^2 .

1.后面的式子提取x^2接下来借好做了
2.直接进行分子有理化,将后面的式子下面除以1,上下同时乘以 (根号（n^2+2n）+根号（n^2-1）)
如果还有疑问,可加QQ：982860636.（为大一学弟讲解）ps：别忘记给分喔~

1.洛比达法则 π/cos^2πx=π
2.=tanx/tan2x=x/2x=1/2
3.洛比达法则 2sinx/-3=-根号3/3
欢迎追问!

1.lim x→0 (x/|x|)
1.lim x→0+ (x/|x|) = 1
lim x→0- (x/|x|) = -1
所以不存在
2.lim x→1 (1/x-1) = 无穷大

lim x->0 (e^x-1)/(2x^2-x)
=lim x->0 x/(2x^2-x) 【等价无穷小代换】
=lim x->0 1/(2x-1)
= -1
lim x->0 2ln[1+(cosx-1)]/x^2
=lim x->0 2(cosx-1)/x^2 【等价无穷小代换】
= -2lim x->0 (1-cosx)/x^2
= -2lim x->0 (x^2 / 2)/x^2
= -1
y=(2x-x^2)^3,
则y'=3(2x-x^2)^2·(2-2x)
y'│x=2
=0
y=2x^5-x^4+x^2-12,
y'=10x^4 - 4x^3 + 2x
y''=40x^3 - 12x^2 + 2
y=ln(x^2-1)+sinx,
y'=2x/(x^2-1) + cosx,
即 dy=[2x/(x^2-1) + cosx]dx

1
lim x→0y→2 sin(xy)/x
=lim x→0y→2 (xy)/x
=lim x→0y→2 y
= 2
2
lim x→0y→0 (2-√（xy+4）)/xy
= lim x→0y→0 (4-（xy+4）)/[(xy)·(2+√（xy+4）)]
= lim x→0y→0 -（xy）/[(xy)·(2+√（xy+4）)]
= lim x→0y→0 -1/[(2+√（xy+4）)]
= -1/4

1.分子分母同时乘以1/n^3
lim n→∞ (n2+5)/(n+1)(n+2)(n+3)
=lim n→∞ (1/n+5/n^3)/(1+1/n)(1+2/n)(1+3/n)
n→∞时,1/n,5/n^3均趋向0.
分子中,1/n,2/n,3/n趋向0
所以原式=0
lim x→0 ((x+sinx)/x
lim x→0 x/x+ lim x→0 sinx/x
=1+1
=2
lim x→0 ((1-cosx)/x^2
=lim x→0 2[sin(x/2)]^2/x^2
=lim x→0 2(x/2)^2/x^2
= 2*(1/2)^2
=1/2
4.y'=(1/x^2)'=(x^(-2))'=-2x^(-3)
5.
e^y-e^x+xy=0
隐函数求导,
对x求导,
d(e^y)/dx-d(e^x)/dx+d(xy)/dx=0
化为：
e^y*dy/dx-e^x+y+xdy/dx=0
则,
dy/dx=(e^x-y)/(e^y+x)