Question

Calcule o valor do limite ou especifique que ele não existe:

lim raiz quadrada de x^2-1 menos raiz quadrada de x+1 e tudo sobre x-3
x-->2

Answer 1

$\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}-\sqrt{x+1}}{x-3}$

Podemos separar o limite em dois limites:

$\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}-\sqrt{x+1}}{x-3}=\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}}{x-3}-\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x+1}}{x-3}\\\\\\\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}-\sqrt{x+1}}{x-3}=\dfrac{\sqrt{2^{2}-1}}{2-3}-\dfrac{\sqrt{2+1}}{2-3}\\\\\\\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}-\sqrt{x+1}}{x-3}=\dfrac{\sqrt{4-1}}{(-1)}-\dfrac{\sqrt{3}}{(-1)}=-\dfrac{\sqrt{3}}{3}+\dfrac{\sqrt{3}}{3}$

Logo:

$\boxed{\boxed{\lim\limits_{x\rightarrow2}~\dfrac{\sqrt{x^{2}-1}-\sqrt{x+1}}{x-3}=0}}$