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PROBLEM:
If \displaystyle f\left(x\right)=x^2-2x+3, find \displaystyle \lim\limits_{x\to 2}\left(\frac{f\left(x\right)-f\left(2\right)}{x-2}\right).
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\begin{align*}
\displaystyle \lim\limits_{x\to 2}\left(\displaystyle \frac{f\left(x\right)-f\left(2\right)}{x-2}\right) & =\lim\limits_{x\to 2}\left(\displaystyle \frac{\left(x^2-2x+3\right)-\left(2^2-2\cdot 2+3\right)}{x-2}\right) \\
\\
& =\lim\limits_{x\to 2}\left(\displaystyle \frac{\left(x^2-2x+3\right)-3}{x-2}\right)\\
\\
& =\lim\limits_{x\to 2}\left(\displaystyle \frac{x^2-2x}{x-2}\right)\\
\\
\end{align*}Direct substitution of x=2 gives the indeterminate form \frac{0}{0}. Therefore, we proceed by factoring the numerator.
\begin{align*}
& =\lim\limits_{x\to 2}\left(\displaystyle \frac{x\left(x-2\right)}{x-2}\right) \\
\\
& =\lim\limits_{x\to 2}\left(x\right) \\
\\
& =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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