Question

$\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right](a\ne 0),$ where $[\cdot ]$ denotes the greatest integer function, is equal to

• A. $a$
• B. $b$
• C. $\frac{b}{a}$
• D. $1-\frac{b}{a}$

Answer 1

The correct option is C $\frac{b}{a}$

$\frac{b}{x}-1<\left[\frac{b}{x}\right]\le \frac{b}{x}$
$\frac{x}{a}(\frac{b}{x}-1)<\frac{x}{a}\left[\frac{b}{x}\right]\le \frac{b}{x}\cdot \frac{x}{a}$
$\underset{x\to 0}{lim}\frac{x}{a}(\frac{b}{x}-1)<\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]\le \underset{x\to 0}{lim}\frac{b}{a}$
Therefore,
$\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]=\frac{b}{a}$