Question

$\underset{x\to \infty }{lim}\frac{\log \left[x\right]}{x}$, where [$x$] denotes the greatest integer less than or equal to $x$, is?

• A. $0$
• B. $1$
• C. $-1$
• D. Non-existent

Answer 1

The correct option is A $0$

Let $x=n+k$, where $0<k<1$. Then, $\left[x\right]=n$

$\therefore \underset{x\to \infty }{lim}\frac{\log \left[x\right]}{x}=\underset{n\to \infty }{lim}\frac{\log n}{n+k}$

$\Rightarrow \underset{x\to \infty }{lim}\frac{\log \left[x\right]}{x}=\underset{n\to \infty }{lim}\frac{1}{n}=0$ [Using L' Hospital's Rule]
Hence, option 'A' is correct.