Part $3$: Practical Application using limit theorem $[30$ Points$]$

Consider the functions $f(n)=n^3$ and $g(n)=10n^2.$ Use the limit theorem to

determine if $f(n)$ is $O(g(n)),(g(n)),$ or $(g(n)).$ Provide detailed steps and

justification for your answer.$[10$ points$]$

Evaluate the relationship between $f(n)=n^2+100n$ and $g(n)=n^{2\mathrm{.}5}$ using the

limit theorem. Determine if $f(n)$ is asymptotically less than, equal to$,$ or greater

than $g(n).$ Show all your workings and provide a thorough justification for your

conclusion.$[10$ points$]$

Given

$f(n)=\frac{n^2+n+1}{n^2-2},$ and $,g(n)=\frac{3n^2-1}{2n^2+5},$

find the limit of

$\frac{f(n)}{g(n)}$

as $n$ tends to infinity. Determine whether $f(n)$ and $g(n)$ have the same asymptotic

behavior.$[10$ points$]$