Question

If $\displaystyle\lim_{x\to \infty}\: x \, \sin \left( \frac{1}{x}\right) = A$ and $\displaystyle\lim_{x\to 0}\: x \, \sin \left( \frac{1}{x}\right) = B$ 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 = \displaystyle\lim_{x\to \infty}\: x \, \sin \left( \frac{1}{x}\right) = \displaystyle\lim_{x\to \infty} \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 = \displaystyle\lim_{t \to \infty} \frac{\sin \, t}{t} = 1$
$\left[ \because \: \displaystyle\lim_{t \to 0} \frac{\sin \, x}{x} = 1 \right] $
and $B = \displaystyle\lim_{x\to 0} \, x \, \sin \left( \frac{1}{x} \right)$
$\Rightarrow \: B = \displaystyle\lim x_{x\to 0} .\displaystyle\lim_{x\to 0} \, \, \sin \left( \frac{1}{x} \right)$
$ \Rightarrow \: B = 0$
$ \therefore $ A = 1 and B = 0 is correct