Question

Limite de ( x³ - 5x²) / 3x , quando x tende a infinito negativo?
Lim ( x³ - 5x²) / 3x
X -> -∞

Answer 1

lim (x^3 - 5x^2)/3x = lim x(x^2-5x)/3x = lim (x^2 - 5x)/3 = (-∞)^2 - (-5∞)/3 = ∞/3 = ∞
Considere que os limites estão todos tendendo a -∞

Answer 2

$L=\underset{x\to -\infty}{\mathrm{\ell im}}~\dfrac{x^3-5x^2}{3x}\\\\\\ =\underset{x\to -\infty}{\mathrm{\ell im}}~\dfrac{x^3\cdot \left(1-\frac{5}{x} \right )}{x\cdot 3}\\\\\\ =\underset{x\to -\infty}{\mathrm{\ell im}}~\dfrac{x^2}{3}\cdot \left(1-\dfrac{5}{x} \right )\\\\\\ =\underset{x\to -\infty}{\mathrm{\ell im}}~f(x)\cdot g(x)~~~~~~\mathbf{(i)}$


onde

$f(x)=\dfrac{x^2}{3}~~\text{ e }~~g(x)=1-\dfrac{5}{x}$

________

Sabemos que

$\underset{x\to -\infty}{\mathrm{\ell im}}~f(x)\\\\ =\underset{x\to -\infty}{\mathrm{\ell im}}~\dfrac{x^2}{3}=+\infty$


e que

$\underset{x\to -\infty}{\mathrm{\ell im}}~g(x)\\\\ =\underset{x\to -\infty}{\mathrm{\ell im}}~\left(1-\dfrac{5}{x} \right )\\\\\\ =1-0\\\\ =1~~~~~(\text{e }1>0)$


Logo, por propriedades operatórias dos limites infinitos,

$\underset{x\to -\infty}{\mathrm{\ell im}}~f(x)\cdot g(x)=+\infty\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \underset{x\to -\infty}{\mathrm{\ell im}}~\dfrac{x^3-5x^2}{3x}=+\infty \end{array}}$


Bons estudos! :-)