Question

Assertion :$\underset{x\to 0^{+}}{lim}x\sin \frac{1}{x}=1$ Reason: $\underset{t\to \infty }{lim}t\sin \frac{1}{t}=1$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

The Assertion (A) is false, as $x\to 0^{+}$ the function $x\sin \frac{1}{x}=$( a finite quantity approaching to zero)$\times$(finite number between 0 & 1)
$\therefore$ $\underset{x\to 0^{+}}{lim}x\sin \left(\frac{1}{x}\right)\to 0$
The Reason (R) is true, since it is nearly equal to the
$\underset{x\to 0}{lim}\frac{\sin x}{x}=1$