Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 2

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PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 2}\left(\frac{x^2+2x-8}{3x-6}\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^2+2x-8}{3x-6}\right)& =\lim\limits_{x\to 2}\left(\frac{\left(x+4\right)\left(x-2\right)}{3\left(x-2\right)}\right)\\
\\
&=\lim\limits_{x\to 2}\left(\frac{x+4}{3}\right)\\
\\
&=\frac{2+4}{3}\\
\\
&=\frac{6}{3}\\
\\
& =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\
\\
\end{align*}

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