Question

Considere a função:

$f(x)= \frac{ x^{2} -5x+6}{x-2}$

Qual é o valor de $\lim_{x \to \infty} f(x)$?

Answer 1

$\underset{x\to \infty}{\mathrm{\ell im}}~\dfrac{x^2-5x+6}{x-2}\\\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~\dfrac{x^2\cdot \left(1-\frac{5}{x}+\frac{6}{x^2}\right)}{x\cdot \left(1-\frac{2}{x} \right )}\\\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~\dfrac{x\cdot \diagup\!\!\!\! x\cdot \left(1-\frac{5}{x}+\frac{6}{x^2}\right)}{\diagup\!\!\!\! x\cdot \left(1-\frac{2}{x} \right )}\\\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~x\cdot \dfrac{1-\frac{5}{x}+\frac{6}{x^2}}{1-\frac{2}{x}}\\\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~g(x)\cdot h(x)$


onde

$g(x)=x~~\text{ e }~~h(x)=\dfrac{1-\frac{5}{x}+\frac{6}{x^2}}{1-\frac{2}{x}}$

_________

Sabemos que

$\underset{x\to \infty}{\mathrm{\ell im}}~g(x)\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~x=\infty$


e que

$\underset{x\to \infty}{\mathrm{\ell im}}~h(x)\\\\ =\underset{x\to \infty}{\mathrm{\ell im}}~\dfrac{1-\frac{5}{x}+\frac{6}{x^2}}{1-\frac{2}{x}}\\\\\\ =\dfrac{1-0+0}{1-0}=1~~~~~(1>0)$


Logo, por propriedades operatórias de limites infinitos, temos que

$\underset{x\to \infty}{\mathrm{\ell im}}~g(x)\cdot h(x)=\infty\\\\\\ \therefore~~\boxed{\begin{array}{c} \underset{x\to \infty}{\mathrm{\ell im}}~\dfrac{x^2-5x+6}{x-2}=\infty \end{array}}$


Bons estudos! :-)