Question

Lim (raiz( 1+x) -1)/-x
x tende a 0

Answer 1

$\lim_{x \to 0} \dfrac{\sqrt{1+x}-1}{-x}$

Racionalize o numerador:

$\lim_{x \to 0} \dfrac{\sqrt{1+x}-1}{-x}\\\\\lim_{x \to 0} \dfrac{\sqrt{1+x}-1}{-x}\cdot \dfrac{\sqrt{1+x}+1}{\sqrt{1+x}+1}\\\\\lim_{x \to 0} \dfrac{1+x-1}{-x-x\sqrt{1+x}}\\\\\lim_{x \to 0} \dfrac{1}{-1-\sqrt{1+x}}$

Por substituição direta:

$\dfrac{1}{-1-\sqrt{1+0}} = \dfrac{1}{-1-\sqrt{1}} = \dfrac{1}{-1-1} = -\dfrac{1}{2}$

Logo:

$\boxed{\lim_{x \to 0} \dfrac{\sqrt{1+x}-1}{-x} = -\dfrac{1}{2}}$