To solve the question, we need to match the correct expressions with their respective limit values. Here is the step-by-step matching:
Given limits and their corresponding results:
1. [tex]\(\lim_{x \rightarrow 6}\left(\frac{x^2-49x-456}{x+8}-\frac{x^2-283x-5,100}{x+17}\right)\)[/tex] matches with [tex]\(243\)[/tex].
2. [tex]\(\lim_{x \rightarrow 4}\left(\frac{\left(x^2+6x-16\right)\left(x^2+16x+63\right)}{x+8}\right)\)[/tex] matches with [tex]\(286\)[/tex].
3. [tex]\(\lim_{x \rightarrow 3}\left(\frac{x^2+99x-202}{x-2}+\frac{x^2+91x-92}{x-1}\right)\)[/tex] matches with [tex]\(199\)[/tex].
4. [tex]\(\lim_{x \rightarrow 7} 21\left(\sqrt{\frac{x^2+187x-3x}{x-2}}\right)\)[/tex] matches with [tex]\(21\sqrt{6685}/5\)[/tex] (not given as an option explicitly here).
5. [tex]\(\lim_{x \rightarrow 2}\left(\frac{\left(x^2+6x+60\right)\left(-x^2-7x+44\right)}{x+6}\right)\)[/tex] matches with [tex]\(247\)[/tex].
6. [tex]\(\lim_{x \rightarrow 8}\left(\frac{20\left(x^2+15x+56\right)}{x+8}-\frac{x^2+3x-10}{x-2}\right)\)[/tex] matches with [tex]\(287\)[/tex].
Therefore, the correct matching according to the calculated results are:
[tex]\[
\begin{array}{cc}
\lim_{x \rightarrow 6}\left(\frac{x^2-49x-456}{x+8}-\frac{x^2-283x-5,100}{x+17}\right) & 243 \\
\lim_{x \rightarrow 4}\left(\frac{\left(x^2+6x-16\right)\left(x^2+16x+63\right)}{x+8}\right) & 286 \\
\lim_{x \rightarrow 3}\left(\frac{x^2+99x-202}{x-2}+\frac{x^2+91x-92}{x-1}\right) & 199 \\
\lim_{x \rightarrow 2}\left(\frac{\left(x^2+6x+60\right)\left(-x^2-7x+44\right)}{x+6}\right) & 247 \\
\lim_{x \rightarrow 8}\left(\frac{20\left(x^2+15x+56\right)}{x+8}-\frac{x^2+3x-10}{x-2}\right) & 287 \\
\end{array}
\][/tex]
Thus, these are the matched pairs based on the provided results.
