# Thread: problem involving limit and greatest integer function

1. ## problem involving limit and greatest integer function

Let f(x) be the greatest integer less than or equal to x.
compute the lim x->0 xf(1/x).

2. It seems the function,
$\displaystyle h(x)=x \left[ \frac{1}{x} \right]$
Is squeeze between,
$\displaystyle f(x)=1, g(x)=1-x$.
On some open interval except possibly containg the point.

We have,
$\displaystyle 1-x \leq h(x)\leq 1$ for $\displaystyle x>0$
$\displaystyle 1\leq h(x)\leq 1-x$ for $\displaystyle x<0$
And,
$\displaystyle \lim_{x\to 0}1-x=\lim_{x\to 0}1=1$
Now just use the squeeze theorem.

(You should graph this function. It is the coolest looking thing).

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# limits involving gif

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