Question

The $\underset{x\to 0}{lim}{x}^8\left[\frac{1}{x^3}\right]$ (where [x] is greatest integer function) is

• A. a nonzero real number
• B. a rational number
• C. an integer
• D. zero

Answer 1

Answer: (B), (C), (D)

The correct options are
B a rational number
C an integer
D zero

Since $x-1\le \left[x\right]\le x$ for all $x\in R$

$\frac{1}{x^3}-1\le \left[\frac{1}{x^3}\right]\le \frac{1}{x^3}$

$\Rightarrow x^8(\frac{1}{x^3}-1)\le x^8\left[\frac{1}{x^3}\right]\le x^5$ for all But

$\Rightarrow \underset{x\to 0}{lim}{x}^5=0=\underset{x\to 0}{lim}(x^5-x^8)$

$\Rightarrow \underset{x\to 0}{lim}{x}^8\left[\frac{1}{x^3}\right]=0\in I\subseteq Q$