a: \(\lim\limits_{x\rightarrow1}\dfrac{\sqrt[3]{x+7}-\sqrt{5-x^2}}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\sqrt[3]{x+7}-2+2-\sqrt{5-x^2}}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\dfrac{x+7-8}{\sqrt[3]{\left(x+7\right)^2}+2\cdot\sqrt[3]{x+7}+4}+\dfrac{4-5+x^2}{2+\sqrt{5-x^2}}}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\dfrac{x-1}{\sqrt[3]{\left(x+7\right)^2}+2\cdot\sqrt[3]{x+7}+4}+\dfrac{x^2-1}{2+\sqrt{5-x^2}}}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(\dfrac{1}{\sqrt[3]{\left(x+7\right)^2}+2\cdot\sqrt[3]{\left(x+7\right)}+4}+\dfrac{x+1}{2+\sqrt{5-x^2}}\right)}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{1}{\sqrt[3]{\left(x+7\right)^2}+2\cdot\sqrt[3]{x+7}+4}+\dfrac{x+1}{2+\sqrt{5-x^2}}\)
\(=\dfrac{1}{\sqrt[3]{\left(1+7\right)^2}+2\cdot\sqrt[3]{1+7}+4}+\dfrac{1+1}{2+\sqrt{5-1^2}}\)
\(=\dfrac{1}{4+2\cdot2+4}+\dfrac{2}{2+2}\)
\(=\dfrac{1}{12}+\dfrac{1}{2}=\dfrac{7}{12}\)
b: \(\lim\limits_{x\rightarrow5}\dfrac{x-5}{\sqrt{x}-\sqrt{5}}\)
\(=\lim\limits_{x\rightarrow5}\dfrac{\left(\sqrt{x}-\sqrt{5}\right)\left(\sqrt{x}+\sqrt{5}\right)}{\sqrt{x}-\sqrt{5}}\)
\(=\lim\limits_{x\rightarrow5}\sqrt{x}+\sqrt{5}=\sqrt{5}+\sqrt{5}=2\sqrt{5}\)