若函数$f(x)=2\sin\left(\dfrac{x}{2}+\dfrac{\pi}{3}\right)$在区间$[-a,a](a>0)$上单调递增，则$a$的取值范围为（ ）
$\text{A.}$ $\left(0,\dfrac{\pi}{3}\right]$
$\text{B.}$ $\left(0,\dfrac{2\pi}{3}\right]$
$\text{C.}$ $\left(0,\dfrac{4\pi}{3}\right]$
$\text{D.}$ $\left(0,\dfrac{5\pi}{3}\right]$