$\underset{x→∞}{\lim}\frac{\log x^n-[x]}{[x]},n∈N$, where [x] denotes the integral part of x, is equal to
Answer & explanation
Correct answer: option 3
$\underset{x→∞}{\lim}\frac{\log x^n-[x]}{[x]}=\underset{x→∞}{\lim}\frac{n\log x-[x]}{[x]}=\underset{x→∞}{\lim}\frac{n\frac{\log x}{x}-\frac{[x]}{x}}{\frac{[x]}{x}}$ …(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 = $\frac{0-1}{1}=-1$