Question

If \(\displaystyle \lim_{x\to\infty}x\sin\!\left(\frac1x\right)=A\) and \(\displaystyle \lim_{x\to0}x\sin\!\left(\frac1x\right)=B\), 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 \(B=1\)

Hint:

Use standard limit \(\displaystyle \lim_{u\to0}\frac{\sin u}{u}=1\).

Answer 1

The Correct Option is A

Step 1: \[ \lim_{x\to\infty}x\sin\!\left(\frac1x\right) =\lim_{u\to0}\frac{\sin u}{u}=1. \]
Step 2: \[ \lim_{x\to0}x\sin\!\left(\frac1x\right)=0 \] since \(|\sin(1/x)|\le1\).