Given $y=(1+\frac{16}{x^2}{)}^{3x^2}$ :

$($a$)$ By hand, rewrite $lny$ as a fraction $\frac{f(x)}{g(x)}.$ Define $f$ and $g$ in Python.

$($b$)$ Find the limits of $f$ and $g$ as $x.$

$($c$)$ If the answers in part $($b$)$ allow for it$,$ use L$\text{'}$H$$pital$\text{'}$s Rule to compute $\underset{x}{lim}y($print all intermediate computations as well$).$

$($d$)$ Evaluate the limit directly using Python to verify your answer to part $($c$).$