Question

lim(x→∞)1/[n^2sin^2(1/3n)]答案是等于9，但这题的解题过程是怎样的？麻烦帮忙解答下，最好详细点~谢谢

Answer 1

利用lim(x→0)sinx/x=1即lim(x→∞)x(sin(1/x))=1
因为n^2sin^2(1/3n)=1/9[1/3n(sin1/3n)]²
所以lim(x→∞)1/[n^2sin^2(1/3n)]=lim(x→∞)1/(1/9[1/3n(sin1/3n)]²)=9

Answer 2

设 x = 1/(3n)
lim(n→∞) 1/[n²sin²(1/3n)]
= lim(x→0) 1/[(1/3x)²sin²x]
= lim(x→0) 1/[1/(9x²)sin²x]
= lim(x→0) 1/[1/9 * sin²x / x² ]
= lim(x→0) 1/[1/9 * (sinx / x)² ]
= 1/[1/9 * 1² ]
= 1 / [1/9]
= 9

Answer 3

lim(x→∞)1/[n^2sin^2(1/3n)]
=lim(x→∞)1/[n^2*(1/3n)^2]
=9

追问

为什么sin能直接去掉呢？

追答

n→∞
1/n→0
用等价无穷小代换即可

Answer 4

lim(x→∞)1/[n^2sin^2(1/3n)]
=lim(x→∞)9{【1/(3n)】/sin(1/3n)}
=9