<s>[INST] Determine the values of $a,b,c$ for which the function is continuous at $x=0$ 
Determine the values of $a,b,c$ for which the function is continuous at $x=0$
  $$
f(x) = 
\begin{cases}
\frac{\sin(a+1)x+sinx}{x} \qquad \text{if} \ x<0 \ ; \\ \\
c \qquad\quad\qquad \text{if} \ x=0 \ ; \\ \\
\frac{\sqrt{x+bx^2}-\sqrt{x}}{bx^{3/2}} \ \ \ \text{if} \ x>0 \ .
\end{cases}
$$

I tried to solve the problem like this:
$$\lim_{x\to 0}\frac{\sin(a+1)x+sinx}{x}$$
$$=\lim_{x\to 0}\frac{acosx}{x}+cosax +1$$
$$=2+a$$
and
$$\lim_{x\to 0}\frac{\sqrt{x+bx^2}-\sqrt{x}}{bx^{3/2}}$$
$$\lim_{x\to 0}\frac{\sqrt{1+bx}-1}{bx}$$
$$=?$$
Can you help me solve this problem?
 [/INST] Hint: $$\frac{\sqrt{1+bx}-1}{bx}=\frac{\sqrt{1+bx}-1}{bx}\cdot\frac{\sqrt{1+bx}+1}{\sqrt{1+bx}+1}=\frac{1}{\sqrt{1+bx}+1}$$
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