Question

The value of $\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]$ is, $\left(\right[.]\to G.I.F)$

• A. $0$
• B. $\infty$
• C. $\frac{b}{a}$
• D. does not exist

Answer 1

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

$\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]$
$=\underset{x\to 0}{lim}\frac{x}{a}(\frac{b}{x}-\left\{\frac{b}{x}\right\})$
Since $\left\{\frac{b}{x}\right\}ϵ[0,1)$
$=\underset{x\to 0}{lim}\frac{x}{a}.\left\{\frac{b}{x}\right\}=0$
$\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]=\underset{x\to 0}{lim}\left(\frac{x}{a}\right)\left(\frac{b}{x}\right)$
$=\underset{x\to 0}{lim}.\frac{b}{a}$
$=\frac{b}{a}$