Question

AJUDA COM ESSE LIMITE, PLEASE!!

$\lim_{x \to \ 0 } \frac{ \sqrt{x + 5,5} - \sqrt{5,5} }{ \sqrt{2x} }$

Answer 1

Temos o seguinte:

$\lim_{x \to 0} \dfrac{ \sqrt{x + 5,5} - \sqrt{5,5} }{ \sqrt{2x} } = \lim_{x \to 0} \dfrac{ \sqrt{x + \frac{11}{2} } - \sqrt{ \frac{11}{2} } }{ \sqrt{2x} }$

Simplificando:

$\lim_{x \to 0} \left (\dfrac{ \sqrt{x + \frac{11}{2} } - \sqrt{ \frac{11}{2} } }{ \sqrt{2x} }\right )* \left ( \dfrac{ \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } }{ \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } } \right) \\ \\ \\ \lim_{x \to 0} \dfrac{ \sqrt{x + \frac{11}{2} }^2 - \sqrt{ \frac{11}{2} }^2 }{ (\sqrt{2x})*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )}$

$\lim_{x \to 0} \dfrac{x + \frac{11}{2} - \frac{11}{2} }{(\sqrt{2x})*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )} \\ \\ \\ \lim_{x \to 0} \left (\dfrac{x }{(\sqrt{2x})*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )} \right) * \dfrac{ \sqrt{2x} }{ \sqrt{2x} } \\ \\ \\ \lim_{x \to 0} \dfrac{x* \sqrt{2x} }{2x*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )}$

$\lim_{x \to0} \dfrac{ \sqrt{2x} }{2*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )}$

Aplicando o limite:

$\lim_{x \to0} \dfrac{ \sqrt{2x} }{2*( \sqrt{x + \frac{11}{2} } + \sqrt{ \frac{11}{2} } )} = \dfrac{ \sqrt{2*0} }{2*( \sqrt{0+ \frac{11}{2} } + \sqrt{ \frac{11}{2} } )} = \dfrac{0}{4 \sqrt{ \frac{11}{2} } } = \boxed{0}$