Consider the piecewise$-$defined function $f(x)$ below. For what value of a does $\underset{x2}{lim}f(x)$ exist?

$f(x)=\{\begin{array}{cc}{x}^2+ax+1,& x<mn>2</mn>\\ 4,& x=2\\ \sqrt[\phantom{2}]{2x+a},& x>2\end{array}$

$($A$)-\frac{7}{4}$

$($B$)-\frac{1}{2}$

$($C$)\frac{3}{4}$

$($D$)12$

$($E$)$ No such value exists