[Math] Determine values of the constants a & b so the limit exists and is equal to f(2) in a piece-wise function

Question

I am trying to determine the values of the constants a & b in a piece-wise function that has to satisfy these parameters:
The limit f(x) as x approaches 2 does exist and is equal to f(2)

The piece-wise part is:

$$
f(x)= {
a+bx if x>2
3 if x=2
b-ax^2 if x<2$$

I have tried solving individually for both a and b, and can come up with a number of solutions that work for each part, but not for both parts.

Basically I know I need to get both functions to pass through the point (2,3) but I really don't know how to go about solving this. I tried using a substitution method, but I don't think I am doing it right as I end up with:

a=3
b=1

neither function passes through (2,3) with these. I'm just not sure how else to approach this

Answer 1

Just solve the following equations $a+2b=3$ and $b-4a=3$ The solution is $a=\frac{-1}{3}$ and $b=\frac{5}{3}$