The function $f:R\to R$ defined by $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{\cos \left(2\pi x\right)-x^{2n}\sin (x-1)}{1+x^{2n+1}-x^{2n}}$  (Where  $n\in N$)  is continuous for all x in :

Let n be a natural number.
$f\left(x\right)=\{\begin{array}{l}\frac{-\sin (x-1)}{x-1}x<-1\\ \cos 2\pi x-1<x<1\\ \frac{-\sin (x-1)}{x-1}x>1\end{array}$
f(x) is discontinuous at x=-2 and x=1