1: \(\lim\limits_{x\rightarrow4}\dfrac{\sqrt{2x+1}-\sqrt{x+5}}{x-4}\)
\(=\lim\limits_{x\rightarrow4}\dfrac{2x+1-x-5}{\sqrt{2x+1}+\sqrt{x+5}}\cdot\dfrac{1}{x-4}\)
\(=\lim\limits_{x\rightarrow4}\dfrac{x-4}{x-4}\cdot\dfrac{1}{\sqrt{2x+1}+\sqrt{x+5}}\)
\(=\lim\limits_{x\rightarrow4}\dfrac{1}{\sqrt{2x+1}+\sqrt{x+5}}=\dfrac{1}{\sqrt{2\cdot4+1}+\sqrt{4+5}}\)
\(=\dfrac{1}{\sqrt{9}+\sqrt{9}}=\dfrac{1}{6}\)
2: \(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1-x}-\sqrt{1+x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{1-x-1-x}{\sqrt{1-x}+\sqrt{1+x}}\cdot\dfrac{1}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{-2x}{x\cdot\left(\sqrt{1-x}+\sqrt{1+x}\right)}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{-2}{\sqrt{1-x}+\sqrt{1+x}}=\dfrac{-2}{\sqrt{1-0}+\sqrt{1+0}}\)
\(=\dfrac{-2}{1+1}=-1\)
