Question

Sabe-se que
$lim \: x \: a( \frac{g(x) ^{2} + f(x) }{3}) = 5$
... por favor confira o print e me ajude a responder ​

Answer 1

Vamos usar as propriedades :

$\displaystyle \lim_{\text x \to \text a} [\ \text{g(x)+f(x)}\ ] = \lim_{\text x \to \text a} \text{g(x)} + \lim_{\text x \to \text a} \text{f(x)} \\\\\\ \lim_{\text x \to \text a} [\text{g(x)}]^{\text n} = [\ \lim_{\text x \to \text a} \text{g(x)} \ ] ^{\text n } \\\\\\ \lim_{\text x \to \text a} \ [\ \text k.\text{g(x)}\ ] = \text k.[ \ \lim_{\text x \to \text a} \text{g(x)} \ ]$

Temos :

$\displaystyle \lim_{\text x \to \text a} [\ \frac{\text{g(x)}^2+\text{f(x)}}{3} \ ] = 5$

Usando a propriedade :

$\displaystyle \frac{1}{3}.[\ \lim_{\text x \to \text a} \ \text{g(x)}^2+ \lim_{\text x \to \text a} \text{f(x)} \ ] =5$  

$\displaystyle [\ \lim_{\text x \to \text a} \ \text{g(x)} \ ]^2+ \lim_{\text x \to \text a} \text{f(x)} =15$

A questão nos diz que :

$\displaystyle \lim_{\text x \to \text a} [\text{g(x)}]^3 = 27$

logo :

$\displaystyle [\ \lim_{\text x \to \text a} \text{g(x)}\ ]^3 = 27 \\\\ \lim_{\text x \to \text a} \text{g(x)} = \sqrt[3]{27} \\\\ \lim_{\text x \to \text a} \text{g(x)} = 3$

Substituindo :

$\displaystyle [\ \lim_{\text x \to \text a} \ \text{g(x)} \ ]^2+ \lim_{\text x \to \text a} \text{f(x)} =15$

$\displaystyle [\ 3\ ]^2+ \lim_{\text x \to \text a} \text{f(x)} =15 \\\\ \lim_{\text x \to \text a} \text{f(x)} = 15 - 9 \\\\ \lim_{\text x \to \text a} \text{f(x)} = 6$

Portanto, concluímos que :

$\huge\boxed{\ \lim_{\text x \to \text a} \text{f(x)}=6 \ ; \ \lim_{\text x \to \text a} \text{g(x)} = 3 \ }\checkmark$