题目

数学题极限题5道
1.lim n→∽ (n2+5)/(n+1)(n+2)(n+3)
2.lim x→0 ((x+sinx)/x,x→0
3.lim x→0 ((1-cosx)/x2,x→0
4.设函数y=1/x2,求y'
5.设函数y=y（x）由方程e的y次方-e的x次方+xy=0,求dy/dx (e的x次方-y)/x+e的y次方

提问时间：2021-02-12

答案

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)