数学题极限题5道

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

4y'=(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^ydy/dx-e^x+y+xdy/dx=0

则，

dy/dx=(e^x-y)/(e^y+x)

lim(1^p+2^p+3^p++n^p)/n^(p+1)

=lim((1/n)^p+(2/n)^p+(3/n)^p++(n/n)^p)1/n

=∫(0,1)x^pdx

=1/(p+1)x^(p+1)|(0,1)

=1/(p+1)