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

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

If $$f(x)=\left\{\begin{array}{cc}\frac{e^{\alpha x}-e^x-x}{x^2}, & x \neq 0 \\ \frac{3}{2}, & x=0\end{array}\right.$$

Find the value of $$\alpha$$ for which the function $$f$$ is continuous