如何证明3n+2/2n+1的极限等于3/2n趋近于无穷大

如何证明3n+2/2n+1的极限等于3/2n趋近于无穷大
如何证明3n+2/2n+1的极限等于3/2
n趋近于无穷大

如何证明3n+2/2n+1的极限等于3/2n趋近于无穷大
lim(3n+2)/(2n+1)
=lim[(3n+3/2)/2*(n+1/2) +1/2*(2n+1)]
=lim[3/2+1/(4n+2)]
=3/2+lim[1/(4n+2)]
n趋近于无穷大,lim[1/(4n+2)]=0
所以：n趋近于无穷大
lim(3n+2)/(2n+1)＝3/2
或者：
lim (3n+2)/(2n+1)
=lim[(3+2/n)/(2+1/n)]
n趋近于无穷大 2/n和1/n均趋向于0
所以：
n趋近于无穷大
lim (3n+2)/(2n+1)
=lim[(3+2/n)/(2+1/n)]＝3/2

lim (3n+2)/(2n+1)
=lim [3+(2/n)]/[2+(1/n)]
=[lim 3+(2/n)]/[lim 2+(1/n)]
=[3+lim (2/n)]/[2+lim (1/n)]
=(3+0)/(2+0)
=3/2
注：n趋于无穷大时，lim 1/n = 0

3n+2/2n+1
=(3n+3/2+1/2)/(2n+1)
=(3n+3/2)/(2n+1)+(1/2)/(2n+1)
=3/2+(1/2)/(2n+1)
后一项极限为0
所以极限为3/2

证明：3n+2/2n+1=【3/2(2n+1）+1/2】/2n+1=3/2+1/4n+2
当n趋近于无穷大，由于1/4n+2无限趋近于0，所以
3n+2/2n+1的极限等于3/2