$($b$)(7$ points$)$ Indicate and justify your answer for the following questions

whether $3^{\frac{n}{2}+6\sqrt[\phantom{2}]{n}+10}=O(2^n)$

whether $3^{\frac{n}{2}+6\sqrt[\phantom{2}]{n}+10}=(2^n)$

whether $3^{\frac{n}{2}+6\sqrt[\phantom{2}]{n}+10}=(2^n)$

using either of the two equivalent asymptotic definitions.

$1$

$($i$)(7$ points$)$ Please answer the above three questions and justify your answers based on the formal definitions of the $O-$notation, $,$ and $-$notation from the lecture notes.

$($Note: For example, to prove $2n=O(n^2),$ let $c=4$ and $n_0=1,$ for every $n>n_0=1,$ we have $2n-4*n^{2}0$ and thus $2nc*n^2.)$

$($ii$)(7$ points$)$ Please answer the above three questions and justify your answers based on the Limit analysis: analyzing the result of $\underset{n}{lim}\frac{3^{\frac{n}{2}+6\sqrt[\phantom{2}]{n}+10}}{2^n}.$

$($Note: The limit analysis of the asymptotic relationship between $f(n)$ and $g(n)$ :

If $\underset{n}{lim}\frac{f(n)}{g(n)},$ then $f(n)=O(g(n)).$

If $\underset{n}{lim}\frac{f(n)}{g(n)}0,$ then $f(n)=(g(n)).$

If $\underset{n}{lim}\frac{f(n)}{g(n)}0$ and $\underset{n}{lim}\frac{f(n)}{g(n)},$ then $f(n)=(g(n)).)$

If $\underset{n}{lim}\frac{f(n)}{g(n)}$ does not exist, then $f(n)$ and $g(n)$ have no asymptotic relationships.