Question

Assertion :If a and b are positive and [x] denotes the greatest integer $\le x$, then $\underset{x\to 0^{+}}{lim}\frac{x}{a}\left[\frac{b}{x}\right]=\frac{b}{a}$ Reason: $\underset{x\to \infty }{lim}\frac{\left\{x\right\}}{x}=0$, where denotes fractional part of x.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right]\underset{x\to 0^{+}}{lim}\frac{x}{a}(\frac{b}{x}-\left\{\frac{b}{x}\right\})=\frac{b}{a}$
$=\underset{x\to 0^{+}}{lim}(\frac{b}{a}-\frac{x}{a}\left\{\frac{b}{x}\right\})$
$=\frac{b}{a}$
Since, $0\le \left\{x\right\}<1$ so $\frac{\left\{x\right\}}{x}\le \frac{1}{x}$ for $x>0$
Hence $x\to \infty \frac{\left\{x\right\}}{x}=0$