$$ \mathop {\lim }\limits_{x \to 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 3 x)(\operatorname{cosec} x-\cot x)^2}= $$

Match the functions in Column I with their properties in Column II. In the following [ $x$ ] denotes the greatest integer less than or equal to $x$.

Consider the following functions

I. $f(x)= \begin{cases}\frac{1}{2}-x & , x<\frac{1}{2} \\ \left(\frac{1}{2}-x\right)^2 & , x \geq \frac{1}{2}\end{cases}$

II. $f(x)=|3 x-1|$

III. $f(x)=x|x|$

IV. $f(x)=|x|$

Then, on $[0,1]$ Lagrange's mean value theorem is applicable to the functions

$$ \mathop {\lim }\limits_{x \to \infty }\left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\ldots+\frac{n+2 n}{n^2+4 n^2}\right]= $$