The function \( f:\mathbb{R}\to \mathbb{R} \) defined by \( f(x)=\displaystyle\lim_{n\to\infty}\frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}} \) is continuous for all \( x \) in:
\( \mathbb{R}-\{-1\} \)
\( \mathbb{R}-\{-1,\,1\} \)
\( \mathbb{R}-\{1\} \)
\( \mathbb{R}-\{0\} \)
