The value of p and q for which the function `f(x)=[((sin(p+1)x+sinx)/x , x<0) , (q , x=0) , ((sqrt(x+x^2)-sqrt(x))/(x^(3//2)), x >0)]` is continuous for all x in R , are : (1) `p=1/2, q=-3/2`
(2) `p=5/2, q=-1/2`
(3) `p=-3/2, q=1/2`
(4) `p=1/2, q=3/2`

Explanation

To determine the values of \( p \) and \( q \) for which the function

\[
f(x) = \begin{cases}
\frac{\sin(p+1)x + \sin x}{x} & \text{if } x < 0 \\
q & \text{if } x = 0 \\
\frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} & \text{if } x > 0
\end{cases}
...