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PROBLEM:
Evaluate \displaystyle \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)
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SOLUTION:
A straight substitution of x=4 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 4}\left(\frac{x^3-64}{x^2-16}\right)& =\lim\limits_{x\to 4}\left(\frac{\left(x-4\right)\left(x^2+4x+16\right)}{\left(x+4\right)\left(x-4\right)}\right)\\
\\
& =\lim\limits_{x\to 4}\left(\frac{x^2+4x+16}{x+4}\right)\\
\\
& =\frac{\left(4\right)^2+4\left(4\right)+16}{4+4}\\
\\
& =\frac{48}{8}\\
\\
& =6 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\
\\
\end{align*}
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