$lim x \to 0 \frac{[x]}{x}$ does not exist as the function is not defined at $x = 0$ , where $[.]$ denotes greatest integer function.
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Solution

At $x = 0$ $[x] = 0$
$∴ lim x \to 0 \frac{[x]}{x} = \frac{0}{0}$ not defined

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