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Determine the values \( a, b, c \), for which the function
\( f(x)=\left\{\begin{array}{ll}\frac{\sin (a+1) x+\sin x}{x}, & \text { for } x<0 \\ c, & \text { for } \quad x=0 \\ \frac{\left(x+b x^{2}\right)^{1 / 2}-x^{1 / 2}}{b x^{3 / 2}}, & \text { for } x>0\end{array}\right. \) is continuous at \( x=0 \).
 
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