Ensino Superior ⇒ Limites no Infinito

[gabriel2153]

[tex3]\lim_{x\to + \infty} (x- \sqrt[3]{2+3x^3})[/tex3]

[candre]

temos:
[tex3]\lim_{x\to+\infty}\left(x-\sqrt[3]{3x^3+2}\right)=\lim_{x\to+\infty}\left[x-\sqrt[3]{x^3\left(3+\cancelto{0}{\frac{2}{x^3}}~~\right)}\right]=\lim_{x\to+\infty}\left(x-\sqrt[3]{3x^3}\right)=\\
\lim_{x\to+\infty}\left(x-\sqrt[3]{3}x\right)=\lim_{x\to+\infty}\left(1-\sqrt[3]{3}\right)x=(1-\sqrt[3]{3})\lim_{x\to+\infty}x=(1-\sqrt[3]{3})\infty\\
\sqrt[3]{3}>1\Rightarrow 1-\sqrt[3]{3}<0\therefore(1-\sqrt[3]{3})\infty=-\infty[/tex3]
resposta:
[tex3]-\infty[/tex3]