If [x] denotes the integral part of x, then $\underset{x→∞}{\lim}\frac{\log_e[x]}{x}=$

0

$x-1<[x]≤x$

$⇒\log_e(x-1)<\log_e[x]≤\log_ex⇒\underset{x→∞}{\lim}\frac{\log_e(x-1)}{x}≤\underset{x→∞}{\lim}\frac{\log_e[x]}{x}≤\underset{x→∞}{\lim}\frac{\log_e x}{x}$

$⇒\underset{x→∞}{\lim}\frac{\frac{1}{x-1}}{1}≤\underset{x→∞}{\lim}\frac{\log_e[x]}{x}≤\frac{1/x}{1}⇒0≤\underset{x→∞}{\lim}\frac{\log_e[x]}{x}≤0⇒\underset{x→∞}{\lim}\frac{\log_e[x]}{x}=0$