Question

Gente me ajudem como resolver esses limites por favor !!

1º) Calcule os limites abaixo:

a) $\lim _{x\to -9}\left(\sqrt{-x-x-10}\right)$

b) $\lim _{x\to -1}\sqrt{2-x^2}$

c) $\lim _{x\to 9}\left(\frac{x\sqrt{x}}{x^2-1}\right)$

d) $\lim _{x\to 3}\left(1-\frac{\sqrt{1+x}}{\sqrt{x-1-x}}\right)$

e) $\lim _{x\to 3}\frac{x-3}{\sqrt{x-\sqrt{3}}\:}$

Answer 1

Boa tarde Luana!
Solução!

O objetivo na resolução de um limite,é eliminar sua indeterminação.

Em alguns casos aqui basta substituir para obter o valor do limite.

$A)\\\\ \lim_{x \to -9} \sqrt{-x-x-10} \\\\\ \lim_{x \to -9} \sqrt{-2x-10}\\\\\ \lim_{x \to -9} \sqrt{-2(-9)-10}\\\\\ \lim_{x \to -9} \sqrt{-2(-9)-10}\\\\\ \lim_{x \to -9} \sqrt{18-10}\\\\\ \lim_{x \to -9} \sqrt{8}\\\\\ \lim_{x \to -9} 2 \sqrt{2} \\\\\\\\\\ \boxed{Resposta: \lim_{x \to -9} \sqrt{-x-x-10}=2 \sqrt{2}}$


$B)\\\\ \lim_{x \to -1} \sqrt{2- x^{2} } \\\\\\ \lim_{x \to -1} \sqrt{2- (-1)^{2} } \\\\\\ \lim_{x \to -1} \sqrt{2 -1 } \\\\\\ \lim_{x \to -1} \sqrt{1 } \\\\\\ \lim_{x \to -1} 1\\\\\\\\\\\ \boxed{Resposta: \lim_{x \to -1} \sqrt{2- x^{2} }=1}$

C)\\\\\

$\lim_{x \to 9} \dfrac{x \sqrt{x} }{ x^{2} -1} \\\\\\\\ \lim_{x \to 9} \dfrac{9 \sqrt{9} }{ (9)^{2} -1} \\\\\\ \lim_{x \to 9} \dfrac{9.3}{ 81 -1} \\\\\\\\\ \lim_{x \to 9} \dfrac{27}{ 80} \\\\\\\\\\\ \boxed{Resposta:\lim_{x \to 9} \dfrac{x \sqrt{x} }{ x^{2} -1}= \frac{27}{80}}$

D)\\\\\

$\lim_{x \to 3} \dfrac{1- \sqrt{1+x} }{ \sqrt{-1} }\\\\\\ \lim_{x \to 3} \dfrac{1- \sqrt{1+x} }{ i} }\\\\\\ \lim_{x \to 3} \dfrac{i- \sqrt{1+x} }{ i} }\\\\\\ \lim_{x \to 3} \dfrac{i+ \sqrt{-1-x} }{ i} }\\\\\\ Lembrando~~i=1~~-1=i^{2}$

$\lim_{x \to 3} \dfrac{i+\sqrt{-1-3} }{ i} }\\\\\\\\ \lim_{x \to 3} \dfrac{i+\sqrt{-4} }{ i} }\\\\\\\\ \lim_{x \to 3} \dfrac{i+2i^{2} }{ i} }\\\\\\\\ \lim_{x \to 3} \dfrac{i(1+2i) }{ i} }\\\\\\\\ \lim_{x \to 3} 1+2i\\\\\\\\\\\ \boxed{Resposta:\lim_{x \to 3} \dfrac{1- \sqrt{1+x} }{ \sqrt{x-1-x} }=1+2i}$

$E)\\\\ \lim_{x \to 3} \dfrac{x-3}{ \sqrt{x- \sqrt{3} } } \\\\\\ \lim_{x \to 3} \dfrac{x-3}{ \sqrt{x- (\sqrt{3} )^{2} } } \\\\\\ \lim_{x \to 3} \dfrac{x-3}{ \sqrt{x-3} } \\\\\\ \lim_{x \to 3} \dfrac{3-3}{ \sqrt{3-3} } \\\\\\ \lim_{x \to 3} \dfrac{0}{ \sqrt{0} } \\\\\\ \lim_{x \to 3} \dfrac{0}{ 0 } \\\\\\ \lim_{x \to 3}~~0$

$\boxed{Resposta:\lim_{x \to 3} \dfrac{x-3}{ \sqrt{x- \sqrt{3} } }=0}$

Boa noite!

Bons estudos!