$\displaystyle \lim_ {x \rightarrow \00} (x[1/x])=? $
Hello,
Can you please help me finding the limit above?
the squared brackets mean floor function.
Thanks,
Michael
To make notation easier say $\displaystyle f(x) = x\left\lfloor {\frac{1}{x}} \right\rfloor $.
Look at the limit $\displaystyle \displaystyle\lim _{x \to 0^ + } f(x)$.
For each $\displaystyle n\in\mathbb{Z}^+$ if $\displaystyle x\in\left(\frac{1}{n+1},\frac{1}{n}\right) $ then $\displaystyle n<\frac{1}{x}<n+1$.
So it follows that $\displaystyle \left\lfloor {\frac{1}{x}} \right\rfloor={n} $ which implies that $\displaystyle \frac{n}{{n + 1}} < f(x) < 1$
So what is the limit on the right?
You can do the limit on the left.