Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined as \( f(x) = a\sin\!\left(\dfrac{\pi [x]}{2}\right) + [2-x],\, a \in \mathbb{R} \), where \( [t] \) is the greatest integer less than or equal to \( t \). If \( \displaystyle\lim_{x\to -1} f(x) \) exists, then the value of \( \displaystyle\int\limits_{0}^{4} f(x)\,dx \) is equal to:
