If f(x)=lim_(n->oo)1/(1+nsin^2pix) , find the value of f(x) for all real values of x?

$f \left(x\right) = 1$ when $x$ is an integer and $f \left(x\right) = 0$ otherwise.
If $x$ is an integer ($x = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$), then $\pi x$ is an integer multiple of $\pi$ so $\sin \left(\pi x\right) = 0$. In this case, $\frac{1}{1 + n {\sin}^{2} \left(\pi x\right)} = \frac{1}{1 + 0} = 1$ for all $n$.
If $x$ is not an integer, then $\sin \left(\pi x\right) \ne 0$ so $n {\sin}^{2} \left(\pi x\right) \to \infty$ as $n \to \infty$. In this case, $\frac{1}{1 + n {\sin}^{2} \left(\pi x\right)} \to 0$ as $n \to \infty$.