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

The value of $$k(k > 0)$$, for which the function $$f(x)=\frac{\left(e^x-1\right)^4}{\sin \left(\frac{x^2}{k^2}\right) \log \left(1+\frac{x^2}{2}\right)}$$, where $$x \neq 0$$ and $$f(0)=8$$

If $$f^{\prime \prime}(x)$$ is continuous at $$x=0$$ and $$f^{\prime \prime}(0)=4$$, then find the following value. $$\lim _\limits{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^2}$$ is equal to