$\quad \lim\limits_{x \to 1}\dfrac{\sqrt[3]{x+7}-2}{\sqrt{x}-1}$
$=\lim\limits_{x \to 1}\dfrac{(\sqrt[3]{x+7}-2)[(\sqrt[3]{x+7})^2+\sqrt[3]{x+7}.2+2^2](\sqrt{x}+1)}{(\sqrt{x}-1)(\sqrt{x}+1)[(\sqrt[3]{x+7})^2+\sqrt[3]{x+7}.2+2^2]}$
$=\lim\limits_{x\to 1}\dfrac{(x+7-2^3)(\sqrt{x}+1)}{(x-1)[(\sqrt[3]{x+7})^2+\sqrt[3]{x+7}.2+2^2]}$
$=\lim\limits_{x\to 1}\dfrac{\sqrt{x}+1}{(\sqrt[3]{x+7})^2+\sqrt[3]{x+7}.2+2^2}$
`={1+1}/{2^2+2.2+4}`
`=1/ 6`
