Let f(x)= lim_θ → 0((cos π x-x^(2/(θ)) sin (x-1))/(1+x^(2/(θ))(x-1))), x ∈ R. Consider the following two statements: (I) f(x) is discontinous at x=1.…

Let $\displaystyle f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in \mathbf{R}$. Consider the following two statements :

$\displaystyle f(x)$ is discontinous at $\displaystyle x=1$.

$\displaystyle f(x)$ is continous at $\displaystyle x=-1$.

Then,