Continuity of two oscillatory functions

Let $f_i:\mathbb{R}\to\mathbb{R}$, $i=1,2$, be defined by $$f_1(x)=\begin{cases}\sin\bigl(\frac1x\bigr)+\cos\bigl(\frac1x\bigr),&\text{if }x\neq 0,\ 0,&\text{if }x=0,\end{cases}$$ and $$f_2(x)=\begin{cases}x\bigl(\sin\bigl(\frac1x\bigr)+\cos\bigl(\frac1x\bigr)\bigr),&\text{if }x\neq 0,\ 0,&\text{if }x=0.\end{cases}$$ Then