Let α and β be the real numbers such that

$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $

Then the value of α + β is ___________.

If

$$ \beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x}, $$

then the value of $6 \beta$ is ___________.

Let $$\alpha$$ be a positive real number. Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ and $$g:(\alpha, \infty) \rightarrow \mathbb{R}$$ be the functions defined by

$$ f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)} . $$

Then the value of $$\lim \limits_{x \rightarrow \alpha^{+}} f(g(x))$$ is

Let the functions $$f:( - 1,1) \to R$$ and $$g:( - 1,1) \to ( - 1,1)$$ be defined by $$f(x) = |2x - 1| + |2x + 1|$$ and $$g(x) = x - [x]$$, where [x] denotes the greatest integer less than or equal to x. Let $$f\,o\,g:( - 1,1) \to R$$ be the composite function defined by $$(f\,o\,g)(x) = f(g(x))$$. Suppose c is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT continuous, and suppose d is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT differentiable. Then the value of c + d is ............