Answer
$ \dfrac{1}{3}$
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
In order to to find the limit, we will plug $ a $ into the function and then simplify.
$\lim_\limits{x\to 2} \dfrac{\dfrac{x^2-4}{x^3-8}-\dfrac{1}{4}}{x}=\lim_\limits{x\to 2} \dfrac{(x-2)(x+2)}{(x-2)(x^2+2x+4)}$
or, $= \lim_\limits{x\to 2} \dfrac{x+2}{x^2+2x+4}$
or, $=\dfrac{2+2}{(2)^2+2(2)+4}$
or, $= \dfrac{1}{3}$
