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Find a limit involving floor function

Mathematics Asked on January 14, 2021

I had to find the following limit: $displaystyle{lim_{x to infty}}frac{x}{lfloor x rfloor}$ Where $xinmathbb{R}$ and $f(x)= {lfloor x rfloor}$ denotes the floor function.

This is what I did:

I wrote $x-1leq lfloor x rfloorleq x$. Then $frac{1}{x}leqfrac{1}{lfloor x rfloor}leqfrac{1}{x-1}$. Multiplying by $x$ we get $frac{x}{x}leqfrac{x}{lfloor x rfloor}leqfrac{x}{x-1}$. Since $displaystyle{lim_{x to infty}}frac{x}{x}=displaystyle{lim_{x to infty}}frac{x}{x-1}=1$, by the squeeze theorem we can say that $displaystyle{lim_{x to infty}}frac{x}{lfloor x rfloor}=1$. Is this correct? If not, what should I fix?

3 Answers

How about

$ displaystyle{lim_{x to infty}}frac{x}{lfloor x rfloor} = {lim_{x to infty}}frac{lfloor x rfloor + {x}}{lfloor x rfloor} = 1+{lim_{x to infty}}frac{{x}}{lfloor x rfloor} = 1$

Answered by steven gregory on January 14, 2021

I'd rather convert it to $$lim_{xto infty}dfrac{x}{x-lbrace x rbrace}=lim_{xto infty}dfrac{1}{1-boxed{dfrac{lbrace x rbrace}{x}}}$$ $$dfrac{lbrace x rbrace}{x}to0 $$since $lbrace x rbrace$ is just a number $in[0,1)$

$therefore $ the limit is $boxed{1}$

And yes, your arguement is also correct

Answered by DatBoi on January 14, 2021

It is correct. Slightly easier $lfloor xrfloor = x-{x}$ So $x/lfloor x rfloor= frac{1}{1-{x}/x}$ so the limit is $1$ since $0le{x}<1$

Answered by JCAA on January 14, 2021

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