Extensions by continuity and oscillation

1. No. Along $a_n=\frac1{n\pi}$ the values are $0$; along $b_n=\frac1{\pi/2+2n\pi}$ the values are $1$. Both sequences tend to $0$, so there is no limit.

2. Yes. Since $\bigl|\sin x\sin\frac1x\bigr|\le|\sin x|\to0$, the limit is $0$. The extension is defined by the value $0$ at $x=0$.

3. No. Take $a_n=\frac1{2n\pi}$ and $b_n=\frac1{(2n+1)\pi}$. Both tend to $0$, and $\cos a_n\cos\frac1{a_n}=\cos a_n\to1$ while $\cos b_n\cos\frac1{b_n}=-\cos b_n\to-1$. So there is no limit.

Rule of thumb. An oscillating factor is harmless when it is multiplied by something tending to $0$, and fatal when it is multiplied by something tending to a non-zero number.