Continuity at $x=2$ requires both one-sided limits to equal the specified value $3$. The right branch gives $a+2b=3$ and the left branch gives $b-4a=3$. Solving these two linear equations gives
$$
\boxed{a=-\frac13,\qquad b=\frac53.}
$$
Substitution gives $-1/3+10/3=3$ on the right and $5/3+4/3=3$ on the left. These conditions enforce continuity, without requiring the two one-sided derivatives to agree.


Solution

Past exam of the mathematical methods II course of the University of Oxford / 2013 / 2 / b / Solution

 Solution (source code)