$\underset{x→∞}{\lim}\frac{\log x^n-[x]}{[x]},n∈N$, where [x] denotes the integral part of x, is equal to

-1

$\underset{x→∞}{\lim}\frac{\log x^n-[x]}{[x]}=\underset{x→∞}{\lim}\frac{n\log x-[x]}{[x]}=\underset{x→∞}{\lim}\frac{n\frac{\log x}{x}-\frac{[x]}{x}}{\frac{[x]}{x}}$    …(i)

But $\underset{x→∞}{\lim}\frac{\log x}{x}$  $\left[\frac{∞}{∞}form\right]$

$=\underset{x→∞}{\lim}\frac{1/x}{1}=0$

and $\underset{x→∞}{\lim}\frac{[x]}{x}=\underset{x→∞}{\lim}\frac{x-\{x\}}{x}=\underset{x→∞}{\lim}\left(1-\frac{\{x\}}{x}\right)=1-0=1$

∴ from (i), required limit = $\frac{0-1}{1}=-1$