For all \( x>0 \), let \( y_{1}(x), y_{2}(x), \) and \( y_{3}(x) \) be the functions satisfying \( \dfrac{dy_{1}}{dx}-(\sin x)^{2}y_{1}=0, \; y_{1}(1)=5, \; \dfrac{dy_{2}}{dx}-(\cos x)^{2}y_{2}=0, \; y_{2}(1)=\dfrac{1}{3}, \; \dfrac{dy_{3}}{dx}-\left(\dfrac{2-x^{3}}{x^{3}}\right)y_{3}=0, \; y_{3}(1)=\dfrac{3}{5e}, \) respectively. Then \( \displaystyle \lim_{x\to 0^{+}} \dfrac{y_{1}(x)y_{2}(x)y_{3}(x)+2x}{e^{3x}\sin x} \) is equal to________.
