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Q. The function $f : R \rightarrow R$ defined by $f(x)=\displaystyle\lim _{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}$ is continuous for all $x$ in

JEE MainJEE Main 2022Limits and Derivatives

Solution:

Note: $n$ should be given as a natural number.
$f(x=\begin{cases} \frac{-\sin (x-1)}{x-1} & x< -1 \\ -(\sin 2+1) & x=-1 \\ \cos 2 \pi x & -1< x< 1 \\ 1 & x=1 \\ \frac{-\sin (x-1)}{x-1} & x >1 \end{cases}$
$f(x)$ is discontinuous at $x=-1$ and $x=1$