数列的极限计算:lim[(7n+4)/(5-3n)]=n→∞lim[(2n^2+n-3)/(3n^2+n-2)]=n→∞lim{[(n+3)(n-4)]/[(n-1)(3-2n)]}n→∞lim[1+4+7+…+(3n-2)]/[1+5+9+…+(4n-3)]n→∞

Question

数列的极限
计算:lim[(7n+4)/(5-3n)]= n→∞
lim[(2n^2+n-3)/(3n^2+n-2)]= n→∞
lim{[(n+3)(n-4)]/[(n-1)(3-2n)]} n→∞
lim[1+4+7+…+(3n-2)]/[1+5+9+…+(4n-3)] n→∞

Answer 1

1 lim[(7n+4)/(5-3n)]
上下同除n
=lim (7+4/n)/(5/n-3)
=-7/3
2 lim[(2n^2+n-3)/(3n^2+n-2)]
上下同除n^2
=lim (2+1/n-3/n^2)/(3+1/n-2/n^2)
=2/3
3 lim{[(n+3)(n-4)]/[(n-1)(3-2n)]}
=lim(n^2-n-12)/(-2n^2+5n-3)
上下同除n^2
=lim (1-1/n-12/n^2)/(-2+5/n-3/n^2)
=-1/2
4 1+4+7+…+(3n-2)=n*[1+3n-2]/2=n(3n-1)/2
1+5+9+…+(4n-3)=n*[1+4n-3]/2=n(4n-2)/2
所以lim[1+4+7+…+(3n-2)]/[1+5+9+…+(4n-3)]
=lim (3n-1)/(4n-2)
上下同除n,
=lim (3-1/n)/(4-2/n)
=3/4