If $$f(x)=\frac{\log _e\left(1+x^2(\tan x)\right)}{\sin x^3}, x \neq 0$$ is to be continuous at $$x=0$$, then $$f(0)$$ must be equal to

$$\mathop {\lim }\limits_{n \to \infty } {{n{{(2n + 1)}^2}} \over {(n + 2)({n^2} + 3n - 1)}}$$ is equal to

If the function $$f(x)$$, defined below, is continuous on the interval $$[0,8]$$, then $$f(x)=\left\{\begin{array}{cc}x^2+a x+b & , \quad 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b & , 4 < x \leq 8\end{array}\right.$$

If $$f(x)$$, defined below, is continuous at $$x=4$$, then

$$f(x) = \left\{ {\matrix{ {{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr {a + b} & , & {x = 4} \cr {{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr } } \right.$$