Mind the Gap at Two

A sensor reading is modelled by the two-branch function

$f(x) = \begin{cases} \dfrac{\sqrt{x + 7} - 3}{x - 2}, & x < 2 \\ \dfrac{x^2 + ax - 10}{x - 2}, & x > 2 \end{cases}$

where $a$ is a constant. For almost every choice of $a$ the right-hand branch blows up as $x \to 2^{+}$. Exactly one choice of $a$ keeps that right-hand limit finite.

With that value of $a$ in place, measure the gap at $x = 2$:

$J = \lim_{x \to 2^{+}} f(x) \;-\; \lim_{x \to 2^{-}} f(x)$

Keep the order of subtraction exactly as written. Give $J$ rounded to two decimal places.