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