Answer
L'Hospital's rule applies.
limit = $12$
Work Step by Step
Theorem 11.3 L'Hospital's Rule:
If $f$ and $g$ are differentiable functions such that
substituting $x=a$ in the expression $\displaystyle \frac{f(x)}{g(x)}$ gives
the indeterminate form $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$, then $\displaystyle \lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a} \displaystyle \frac{f^{\prime}(x)}{\mathrm{g}^{\prime}(x)}$.
That is, we can replace $f(x)$ and $g(x)$ with their derivatives and try again to take the limit.
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When x approaches $2$,
the numerator approaches 0,
the denominator approaches 0,
$\displaystyle \frac{f(x)}{g(x)}$ gives the indeterminate form $\displaystyle \frac{0}{0}$,
L'Hospital's rule applies.
$\displaystyle \lim_{x\rightarrow 2}\frac{x^{3}-8}{x-2}=\lim_{x\rightarrow 2} \displaystyle \frac{[x^{3}-8]^{\prime}}{[x-2]^{\prime}}$
$=\displaystyle \lim_{x\rightarrow 2} \displaystyle \frac{3x^{2}-0}{1-0}= $
$=\displaystyle \frac{3(2^{2})}{1}=12 $
