如何证明3n+2/2n+1的极限等于3/2
来源:百度知道 编辑:UC知道 时间:2024/05/26 10:40:31
n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
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