Question

If $\underset{x\to \infty }{\lim }x\sin \left(\frac{1}{x}\right)=A$ and $\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)=B$ then which one of the following is correct?

• A. A =1 and B = 0
• B. A =0 and B = 1
• C. A =0 and B = 0
• D. A =1 and B = 1

Answer 1

Answer: (A)

As given,
$A=\underset{x\to \infty }{\lim }x\sin \left(\frac{1}{x}\right)=\underset{x\to \infty }{\lim }\frac{\sin \left(\frac{1}{x}\right)}{\left(\frac{1}{x}\right)}$
Let $t=\frac{1}{x}$ when $x\to \alpha ,t\to 0$
$\Rightarrow A=\underset{t\to \infty }{\lim }\frac{\sin t}{t}=1$
$\left[\because \underset{t\to 0}{\lim }\frac{\sin x}{x}=1\right]$
and $B=\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)$
$\Rightarrow B=\lim x_{x\to 0}.\underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)$
$\Rightarrow B=0$
$\therefore$ A = 1 and B = 0 is correct