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?

My preferred is to use $\sqrt{1+h}=1+{h\over 2}+o(h)$
In our case, order one is enough and we need first to write $x=2+h$. The fraction transforms as follows
$${3\left(\sqrt{1+{h\over 9}}-1\right)\over 3\left(\sqrt{1+{4h\over 9}+{h^2\over 9}}-{h\over 3}-1\right)}$$
and this simplifies in one step into
$${{h\over 18}+o(h)\over -{h\over 9}+o(h)}=-{1\over 2}+o(1)$$
And the limit is $-{1\over 2}$
