Question

55 pontos!
(Raiz quadrada (x^2 + x + 1) + x)
X-> - ∞

Answer 1

Conteúdo:

• Limites tendendo ao infinito

• Limites

$\large { \sf Temos \: de\: \bf {multiplicar} \sf \: pelo \: \bf {conjugado}\sf \: de}$ $\large { \sf \sqrt{x^2+x+1}+ x : }$

                           $\uparrow$              $\nearrow$

$\huge {\boxed {\blue {\sf \cfrac{x+1}{\sqrt{x^2+x+1}-x } }}}$

$\huge {\text { \blue {\sf =\lim _{x\to \:-\infty \:}\left(\cfrac{x+1}{\sqrt{x^2+x+1}-x}\right)} }}$

$\large { \sf Dividi \: pelo \bf \: denominador \: de \: maior \: pot \^e ncia: }$

$\huge {\boxed {\red {\sf \left(\cfrac{1+\cfrac{1}{x}}{-\sqrt{1+\cfrac{1}{x}+\cfrac{1}{x^2}}-1}\right)}}}$

Propriedades dos Limites:

$\huge {\boxed { \sf \bf \lim _{x\to a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\cfrac{\lim _{x\to a}f\left(x\right)}{\lim _{x\to a}g\left(x\right)},\:\quad \lim _{x\to a}g\left(x\right)\ne 0}}$

                         $\large { \sf \swarrow}$

™ $\large { \sf Com\:exce \c c \~ao\:da\:forma\:indeterminada }$

$\huge {\text{ \sf =\cfrac{\lim _{x\to \:-\infty \:}\left(1+\frac{1}{x}\right)}{\lim _{x\to \:-\infty \:}\left(-\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}-1\right)} }}$

$\huge {\boxed {\sf \lim _{x\to \:-\infty \:}\left(1+\frac{1}{x}\right) = 1 }}$

$\huge {\boxed {\sf \lim _{x\to \:-\infty \:}\left(-\sqrt{1+\cfrac{1}{x}+\cfrac{1}{x^2}}-1\right) = -2 }}$

∴ Resultado Final...

              $\huge {\boxed {\sf \bf -\cfrac{1}{2} }}$

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