$\lim\limits_{x\to 2}\dfrac{ \sqrt[3]{x^2+2x}-1}{x+2}$
$=\dfrac{\sqrt[3]{2^2+2.2}-1}{2+2}$
$=\dfrac{1}{4}$
$\lim\limits_{x\to +\infty} x(\sqrt{x^2+5}-x)$
$=\lim\limits_{x\to +\infty}x.\dfrac{x^2+5-x^2}{\sqrt{x^2+5}+x}$
$=\lim\limits_{x\to +\infty}\dfrac{5x}{\sqrt{x^2+5}+x}$
$=\lim\dfrac{5}{\sqrt{1+\dfrac{5}{x^2}}+1}$
$=\dfrac{5}{2}$