If a function,

$$ f(x)=\left\{\begin{array}{cc} \frac{\sqrt[3]{1+a x^2+b x^3}-\sqrt[3]{1-a x^2-b x^3}}{x^2}, & x<0 \\ 5, & x=0 \\ \frac{\tan 3 x-\sin 3 x}{b x^3}, & x>0 \end{array}\right. $$

is continuous at $x=0$, then the geometric mean of $a$ and $b$ is

$[x]$ denotes the greater integer less than or equal to $x$. If $\{x\}=x-[x]$ and $\lim\limits_{x \rightarrow 0}-\frac{\sin ^{-1}(x+[x])}{2-\{x\}}=\theta$, then $\sin \theta+\cos \theta=$

$$ \mathop {\lim }\limits_{n \to \infty } \frac{1}{n^3} \sum\limits_{k=1}^n k^2 x= $$

Let $f: R \rightarrow R$ be defined by

$$ f(x)=\left\{\begin{array}{cc} a-\frac{\sin [x-1]}{x-1}, & \text { if } x>1 \\ 1, & \text { if } x=1 \\ b-\left[\frac{\sin [x-1]-[x-1]}{([x-1])^3},\right. & \text { if } x<1 \end{array}\right. $$

where $[t]$ denotes the greatest integer less than or equal to $t$. If $f$ is continuous at $x=1$, then $a+b=$