How to find $\lim\limits _{n\to \infty }(n[\frac{5}{n}])$

Question

how to find

• $$\lim _{n\to \infty }(n[\frac{5}{n}])$$
• $$\lim _{n\to \infty }([n]\frac{5}{n})$$

what I did :

$$\frac{5}{n}-1\le[\frac{5}{n}]\le\frac{5}{n}$$

so: $$\lim _{n\to \infty }(n(\frac{5}{n}-1))\le\lim _{n\to \infty }(n[\frac{5}{n}])\le\lim _{n\to \infty }(n\frac{5}{n})$$

we get : $$-\infty\le\lim _{n\to \infty }(n[\frac{5}{n}])\le5$$

so this don't work any other idea how to solve it ?

but for the second one this work

$$n-1\le[n]\le n$$

so: $$\lim _{n\to \infty }((n-1)(\frac{5}{n}))\le\lim _{n\to \infty }([n]\frac{5}{n})\le\lim _{n\to \infty }(n\frac{5}{n})$$

we get : $$5\le\lim _{n\to \infty }(n[\frac{5}{n}])\le5$$

$$\implies \lim _{n\to \infty }(n[\frac{5}{n}]) = 5$$

thanks

Answer 1

for $n>6$ implies $0<\left[\frac5n\right]<1 $ then we have $$\left[\frac5n\right] = 0\implies n\left[\frac5n\right]=0 \implies \lim_{n\to\infty}n\left[\frac5n\right]=0$$

on the other hand We have $[n]=n$ hence the other limit is trivially 5.

Answer 2

Sequence for $$a(n)=n[\frac{5}{n}] \to 5,4,3,4,5,0,0,0,0,0,0,0,0,0,0,\cdots$$ $$a(n)=[n]\frac{5}{n} \to 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,\cdots$$

Answer 3

1) Clearly $\left(n\left[\dfrac{5}{n}\right]\right)=0$ for $n\gt 5$ so the required limit is equal to $0$.

2) On the other hand $[n]\left(\dfrac{5}{n}\right)=\dfrac {[n]}{n}\cdot5$ and $\dfrac {[n]}{n}$ tends clearly to $1$ so the asked limit is equal to $5$.