Question

Lim (x+1)/(³√(2x+3)-1, x-->-1
Resolva fatorando

Answer 1

recuerde...
$a^3-b^3=(a-b)(a^2+ab+b^2)$

entonces

$\sqrt[3]{2x+3}^3-1^3=\left(\sqrt[3]{2x+3}-1\right)\left(\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1\right)$

Luego

$\displaystyle L=\lim\limits_{x\to-1} \frac{x+1}{\sqrt[3]{2x+3}-1} \\ \\ L=\lim\limits_{x\to-1} \frac{x+1}{\sqrt[3]{2x+3}-1}\cdot \frac{\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1}{\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1} \\ \\ L=\lim\limits_{x\to-1} \frac{(x+1)(\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1)}{2x+3-1} \\ \\ L=\lim\limits_{x\to-1} \frac{(x+1)(\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1)}{2(x+1)}\\ \\ L=\lim\limits_{x\to-1} \frac{\sqrt[3]{2x+3}^2+\sqrt[3]{2x+3}+1}{2}$

$\displaystyle L=\frac{\sqrt[3]{2(-1)+3}^2+\sqrt[3]{2(-1)+3}+1}{2} \\ \\ \boxed{L=\frac{3}{2}}$