Identify the function $g$ and value of $a$ such that the following limit represents $g^{\prime }\left(a\right)$. Then, evaluate the limit by finding the value of $g^{\prime }\left(a\right)$.
$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}$

$g\left(x\right)=2x^3$

$a=2$

$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}=18$

$g\(\left\)(x\(\right\))=2x^3$

$a=2$

$\(\displaystyle\) \(\lim\)_{x \(\to\) 2}{\(\frac{2x^3-16}{x-2}\)}=24$

$g\left(x\right)=6x^2$

$a=16$

$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}=18$

$g\left(x\right)=6x^2$

$a=16$

$\underset{x\to 2}{\lim }\frac{2x^3-16}{x-2}=24$