Let $f: R \rightarrow R$ be a function defined by $f(x)=\left\{\begin{array}{l}\frac{\sin \left(x^2\right)}{x} \text { if } x \neq 0 \\ 0 \text { if } x=0\end{array}\right\}$ Then, at $x=0, f$ is

Let $f(\theta ) = \sin \theta (\sin \theta + \sin 3\theta )$, then $f(\theta )$

If $f(x) = \cos (\log x)$, then $f({x^2})f({y^2}) – \frac{1}{2}\left[ {f\,\left( {\frac{{{x^2}}}{2}} \right) + f\left( {\frac{{{x^2}}}{{{y^2}}}} \right)} \right]$ has the value

If $f(x)$ is a polynomial function satisfying the condition $f(x) . f(1/x) = f(x) + f(1/x)$ and $f(2) = 9$ then :

Consider a function $f:\left[ { – 1,1} \right] \to R$ where $f(x) = {\alpha _1}{\sin ^{ – 1}}x + {\alpha _3}\left( {{{\sin }^{ – 1}}{x^3}} \right) + ….. + {\alpha _{(2n + 1)}}{({\sin ^{ – 1}}x)^{(2n + 1)}} – {\cot ^{ – 1}}x$ Where $\alpha _i\ 's$ are positive constants and $n \in N < 100$ , then $f(x)$ is