Solving $\lim_{x\to2^{+}}\frac{\sqrt{x+7}-3}{\sqrt{x^{2}+5}-x-1}$ I need to solve $$\lim_{x\to2^{+}}\frac{\sqrt{x+7}-3}{\sqrt{x^{2}+5}-x-1}$$ without using L'hopital or taylor. tried Conjugate multiplication to no end. Any ideas?

You should give conjugate multiplication a second chance:
$$\begin{align}
\frac{\sqrt{x+7}-3}{\sqrt{x^{2}+5}-x-1}&=\frac{(\sqrt{x+7}-3)(\sqrt{x+7}+3)(\sqrt{x^{2}+5}+x+1)}{(\sqrt{x^{2}+5}-x-1)(\sqrt{x^{2}+5}+x+1)(\sqrt{x+7}+3)}\\
&=\frac{(x-2)(\sqrt{x^{2}+5}+x+1)}{(x^2+5-(x+1)^2)(\sqrt{x+7}+3)}\\
&=-\frac{\sqrt{x^{2}+5}+x+1}{2(\sqrt{x+7}+3)}.
\end{align}$$
