$$ \mathop {\lim }\limits_{x \to \infty } \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}= $$

$$ \mathop {\lim }\limits_{x \to 0} \frac{63^x-9^x-7^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=\ldots \ldots $$

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable function having $\mathrm{f}(3)=3, \mathrm{f}^{\prime}(3)=\frac{1}{27}$ and $\mathrm{g}(x)= \begin{cases}\int_3^{\mathrm{f}(x)} \frac{3 \mathrm{t}^2}{x-3} \mathrm{dt}, & \text { if } x \neq 3 \\ \mathrm{~K}, & \text { if } x=3\end{cases}$ is continuous at $x=3$, then $\mathrm{K}=$

If Rolle's theorem holds for the function $x^3+\mathrm{a} x^2+\mathrm{b} x, 1 \leq x \leq 2$ at the point $\frac{4}{3}$, then the values of $a$ and $b$ are respectively