Question

$\underset{x\to \infty }{lim}\frac{\log x^n-\left[x\right]}{\left[x\right]},n\in N$
($\left[x\right]$ denotes the greatest integer less than or equal to $x$)

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

Answer 1

The correct option is A $-1$

$\underset{x\to \infty }{lim}\frac{\log x^n-\left[x\right]}{\left[x\right]}$

$\underset{x\to \infty }{lim}\frac{n\log x-\left[x\right]}{\left[x\right]}$

$\underset{x\to \infty }{lim}\frac{n\log x}{\left[x\right]}-1$

$n\underset{x\to \infty }{lim}\frac{\log x}{\left[x\right]}-1$

Now we know that as $x\to \infty ,\frac{\log x}{\left[x\right]}\to 0$

$n\left(0\right)-1=-1$