Question

$\underset{x\to \infty }{lim}{x}^2\left(\sqrt{\frac{x+2}{x}}-\sqrt[3]{3\frac{x+3}{x}}\right)$

Answer 1

Given,

${lim}_{x\to \infty }\left(x^2(\sqrt{\frac{x+2}{x}}-\sqrt[3]{3\frac{x+3}{3}})\right)$

${lim}_{x\to \infty }\left(x^2\right)\cdot {lim}_{x\to \infty }(\sqrt{\frac{x+2}{x}}-\sqrt[3]{3\frac{x+3}{3}})$

$=\infty \cdot (1-\infty )$

$=-\infty$

Hence, this is the answer.