If a real valued function
$$ f(x)=\left\{\begin{array}{cl} \frac{x^2+(a+3) x+(a+1)}{x+3} & , \text { when } x \neq-3 \\ -\frac{5}{2} & , \text { when } x=-3 \end{array}\right. $$
is continuous at $x=-3$, then $\lim _{x \rightarrow a}\left(x^2+x+1\right)=$
$$ \mathop {\lim }\limits_{x \to 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 3 x)(\operatorname{cosec} x-\cot x)^2}= $$
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 $x$.
| Column I | Column II | ||
|---|---|---|---|
| A | $$ x|x| $$ |
I | Strictly increasing and continuous in $(-1,1)$ |
| B | $$ \sqrt{|x|} $$ |
II | Continuous but not differentiable in $(-1,1)$ |
| C | $$ x+[x] $$ |
III | Differentiable in $(-1,1)$ |
| D | $$ |x-1|+|x+1|+|x| $$ |
IV | Differentiable in $(-1,0) \cup(0,1)$ |
| V | Strictly increasing and not differentiable in $(-1,1)$ | ||
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
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