Find all values of parameters to make a certain function continuous. $$f(x) = \begin{cases}
\frac{\sin(x^2 - 4)}{x-2} & \text{if $x \in (-\infty, 2)$} \\
ax+b & \text{if $x \in [2,5]$} \\
{\sqrt{4(\ln(x-5))^2 + 1} \over \ln(x-5)} & \text{if $x \in (5, \infty)$}
\end{cases}$$
My process is equating the limit as $x$ approaches $2$ from the left and from the right: 
\begin{gather*}
\lim_{x\to 2^-} \frac{\sin(x^2 - 4)}{x-2} = L = 4\\
\lim_{x\to 2^+} ax+b = L = 2a+b
\end{gather*}
Then $2a+b = 4 \implies b = 4 - 2a$.

\begin{gather*}
\lim_{x\to 5^-} ax+b = -5a+b \\
\lim_{x\to 5^+} {\sqrt{4(\ln(x-5))^2 + 1} \over \ln(x-5)}  = -2
\end{gather*}
Then $-5a+b = -2$

$\therefore -5a+4-2a = -2 $
$\therefore -7a = -6$
$\therefore a = 6/7$
Is this right for getting the $a$ part?

Looks pretty good! You just made a small sign mistake. You should have:
$$
\lim_{x\to 5^-} ax+b = \color{red}{+5}a+b
$$
Hence, by equating the left hand side and right hand side limits and substituting, we obtain:
$$
-2 = 5a + b = 5a + (4-2a) = 3a + 4 \iff -6 = 3a \iff \boxed{a = -2}
$$
