How to set up equation to find the unknown values if a limit exist. 
For what values of the constants $a$ and $b$ does the following limit exist?
  $$\lim_{x\to0}\frac{|x+3|(\sqrt{ax+b}-2)}x$$

for this question, $$f(x) = \frac{-(x+3)(\sqrt{ax+b}-2)}x,x<-3$$
$$f(x) = \frac{(x+3)(\sqrt{ax+b}-2)}x,x>=-3$$
Firstly, I found difficulties in setting up two equation solving for a and b.
Secondly, since $|x+3|=x+3$ only when $x\to0$, I can only use the follwoing to set up the equation. $$\lim_{x\to0}\frac{(x+3)(\sqrt{ax+b}-2)}x$$ 
However, I cannot simplified it to cancel out the denominator $x$ by some method like rationalization. It gets $$\lim_{x\to0}\frac{(x+3)(ax+b-4)}{(\sqrt{ax+b}-2)x}$$.
Therefore, how can we set up the equation to find $a$ and $b$ or it needs other methods to do this type of questions?

$\lim_{x\to0}\frac{|x+3|(\sqrt{ax+b}-2)}x
$
Since
$\lim_{x\to0} |x+3|
=3$,
that doesn't matter.
What is left is
$\lim_{x\to0}\frac{(\sqrt{ax+b}-2)}x
$.
For this limit to exist,
we must have
$\lim_{x\to0}\sqrt{ax+b}-2
=0
$
or
$\lim_{x\to0}\sqrt{ax+b}
=2
$.
Since
$\lim_{x\to0}ax
=0
$,
we must have
$b=4
$.
The answer is,
therefore,
$b=4$
and any value of $a$.
