Question

STATEMENT-1 : If a and b are positive and [x] denotes the greatest integer less than or equal to x, then $\underset{x\to 0^{+}}{lim}\frac{x}{a}\left[\frac{b}{x}\right]=\frac{b}{a}$.
STATEMENT-2 : $\underset{x\to \infty }{lim}\frac{\left\{x\right\}}{x}\to 0$, where {x} denotes the fractional part of x.

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

The correct option is A STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

$L=\underset{x\to 0^{+}}{lim}\frac{x}{a}\left[\frac{b}{x}\right]$
$=\underset{x\to 0^{+}}{lim}\frac{x}{a}(\frac{b}{x}-\left\{\frac{b}{x}\right\})$
$=\underset{x\to 0^{+}}{lim}(\frac{b}{a}-\frac{x}{a}\left\{\frac{b}{x}\right\})$
$=\frac{b}{a}-\frac{b}{a}\underset{x\to 0^{+}}{lim}\frac{\left\{\frac{b}{x}\right\}}{\frac{b}{x}}$
$=\frac{b}{a}-\frac{b}{a}\underset{y\to \infty }{lim}\frac{\left\{y\right\}}{y}$ (where $y=\frac{b}{x}$ and b > 0) ($whenx\to 0,y\to \infty$)
$=\frac{b}{a}$
Also, if $b<0,L=\frac{b}{a}-\frac{b}{a}\underset{y\to -\infty }{lim}\frac{\left\{y\right\}}{y}=\frac{b}{a}$.