Question

Let $f\left(x\right)=x^5\left[1/x^3\right]x\ne 0$ and $f\left(0\right)=0$ , where $\left[x\right]$ denotes greatest integer function, then :

• A. $\underset{x\to 0}{lim}f\left(x\right)$ doesn't exist
• B. f is not continuous at $x=0$
• C. $\underset{x\to 0}{lim}f\left(x\right)=1$
• D. $\underset{x\to 0}{lim}f\left(x\right)=0$

Answer 1

The correct option is D $\underset{x\to 0}{lim}f\left(x\right)=0$

Since $x-1\le \left[x\right]\le x$ for all $x\in R$
$\frac{1}{x^3}-1\le \left[\frac{1}{x^3}\right]\le \frac{1}{x^3}$
$\Rightarrow x^5(\frac{1}{x^3}-1)\le x^5\left[1/x^3\right]\le x^2$ for $x>0$
and $x^2\le x^5\left[1/x^3\right]\le x^5\left(\right(1/x^3)-1)$ for $x<0$
so $\underset{x\to 0}{lim}{x}^5\left[1/x^3\right]=0$