The Taylor series expansion of sin $$x$$ about $$x = {\pi \over 6}$$ is given by

Limit of the function
$$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$ is given by

If $$f\left( {x,y,z} \right) = $$
$${\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}},$$ $${{{\partial ^2}f} \over {\partial {x^2}}} + {{{\partial ^2}f} \over {\partial {y^2}}} + {{{\partial ^2}f} \over {\partial {z^2}}}$$ is equal to _______.

Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.