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  4. displaystyle lim x arrow ∞ ( log [x]/x), where [.] denotes greatest integral function, is

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Q. $\displaystyle\lim _{x \rightarrow \infty} \frac{\log [x]}{x}$, where [.] denotes greatest integral function, is

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \infty} \frac{\log [x]}{x}=\lim _{x \rightarrow \infty} \frac{\log x}{x}$
$(\because$ when $x \rightarrow \infty,[x] \approx x$ )
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\frac{1}{x}}{1}=0$
(Applying L' Hospital's rule)