\(\displaystyle \lim_{x \rightarrow 2} \: \frac{2}{x^2-4}-\frac{3}{x^2+2x-8}\)
\begin{equation*}
\begin{split}
& \lim_{x \rightarrow 2} \: \frac{2}{x^2-4}-\frac{3}{x^2+2x-8} \\\\
& \lim_{x \rightarrow 2} \: \frac{2}{(x - 2)(x + 2)}-\frac{3}{(x - 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{2(x + 4) - 3(x + 2)}{(x - 2)(x + 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{2x + 8 - 3x - 6}{(x - 2)(x + 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{-x + 2}{(x - 2)(x + 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{- (x - 2)}{(x - 2)(x + 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{- \cancel {(x - 2)}}{\cancel {(x - 2)}(x + 2)(x + 4)} \\\\
& \lim_{x \rightarrow 2} \: \frac{-1}{(x + 2)(x + 4)} \\\\
& \frac{-1}{(2 + 2)(2 + 4)} \\\\
& \frac {-1}{4 \:.\: 6} \\\\
& -\frac{1}{24}
\end{split}
\end{equation*}