Question

calcule o limite
lim->0 (raizx+4)-2/2x

Answer 1

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{\sqrt{x + 4} - 2}{2x}$

Fazendo a substituição:

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{\sqrt{0 + 4} - 2}{2\cdot0}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{\sqrt{4} - 2}{0}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{2 - 2}{0}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{0}{0}$

O valor é indeterminado

Calcular a derivada, sendo $\dfrac{d}{dx}$

Numerador

$\sf = \sqrt{x + 4} - 2$

$\sf = \dfrac{1}{2\sqrt{x + 4}}$

Denominador

$\sf = 2x$

$\sf = 2$

Obtemos:

$\sf \dfrac{1}{\dfrac{2\sqrt{x + 4}}{2}}$

$\sf \dfrac{1}{2\sqrt{x + 4} \cdot 2}$

$\sf \dfrac{1}{4\sqrt{x + 4}}$

Fazendo a substituição:

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{1}{4\sqrt{x + 4}}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{1}{4\sqrt{0 + 4}}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{1}{4\sqrt{4}}$

$\sf \underset{x \Rightarrow 0}{lim}~~\dfrac{1}{4 \cdot 2}$

$\sf \underset{x \Rightarrow 0}{lim}~~\boxed{= \dfrac{1}{8}}$