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PROBLEM:
Evaluate \displaystyle \lim\limits_{x\to 2}\left(\frac{x^3-8}{x^2-4\:}\right)
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SOLUTION:
A straight substitution of x=2 leads to the indeterminate form \frac{0}{0} which is meaningless.
Therefore, to evaluate the limit of the given function, we proceed as follows
\begin{align*}
\lim\limits_{x\to 2}\left(\frac{x^3-8}{x^2-4\:}\right) & =\lim\limits_{x\to 2}\left[\frac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x+2\right)\left(x-2\right)}\right] \\
&=\lim\limits_{x\to 2}\left[\frac{\left(x^2+2x+4\right)}{\left(x+2\right)}\right] \\
& =\frac{2^2+2\cdot 2+4}{2+2} \\
& =3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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