Question

The value of $\underset{x\to \infty }{lim}\frac{\log x}{\left[x\right]}$ is?

where [.] denotes the greatest integer function

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

Answer: (A)

$0$

We have,
$x-1<\left[x\right]\le x$ for all $x$ $\in$ R
$\Rightarrow \frac{1}{x}\le \frac{1}{\left[x\right]}<\frac{1}{x-1}$ for all $x$ $\in R-[0,1]$
$\Rightarrow \frac{\log x}{x}\le \frac{\log x}{\left[x\right]}<\frac{\log x}{x-1}$
$[\because \log x>0asx\to \infty ]$
$\Rightarrow \underset{x\to \infty }{lim}\frac{\log x}{x}\le \underset{x\to \infty }{lim}\frac{\log x}{\left[x\right]}<\underset{x\to \infty }{lim}\frac{\log x}{x-1}$
$\Rightarrow \underset{x\to \infty }{lim}\frac{\log x}{\left[x\right]}=0$
$[\because \underset{x\to \infty }{lim}\frac{\log x}{x}=\underset{x\to \infty }{lim}\frac{\log x}{x-1}=0]$
Hence, option 'A' is correct.