Question: If

\[f\left( x \right)=\left\{ \begin{aligned} & \sqrt{2-x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x<2 \\ & {{x}^{3}}+k\left( x+1 \right)\,\,\,\,\,\,x\ge 2 \\ \end{aligned} \right.\]

determine the value of the constant k for which \(\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)\) exists.