Tính giới hạn $\Large \underset{x\rightarrow 0}{lim}\dfrac{\sqrt[3]{x+

Tính giới hạn $\Large \underset{x\rightarrow 0}{lim}\dfrac{\sqrt[3]{x+1}-1}{\sqrt[4]{2x+1}-1}$

• A.

  $\Large +\infty$.

• B.

  $\Large -\infty$.

• C.

  $\Large \dfrac{2}{3}$.

• D.

  0.

$\Large \underset{x\rightarrow 0}{lim}\dfrac{\sqrt[3]{x+1}-1}{\sqrt[4]{2x+1}-1}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{(\sqrt[3]{x+1}-1)\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)(\sqrt[4]{2x+1}+1)}{(\sqrt[4]{2x+1}-1)(\sqrt[4]{2x+1}+1)\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{x(\sqrt[4]{2x+1}+1)}{(\sqrt[2]{2x+1}-1)\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{x(\sqrt[4]{2x+1}+1)(\sqrt[2]{2x+1}+1)}{(\sqrt[2]{2x+1}-1)(\sqrt[2]{2x+1}+1)\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{x(\sqrt[4]{2x+1}+1)(\sqrt[2]{2x+1}+1)}{2x\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{(\sqrt[4]{2x+1}+1)(\sqrt[2]{2x+1}+1)}{2\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x+1}-1\right)}$

$\Large =\underset{x\rightarrow 0}{lim}\dfrac{2.2}{2(1+1+1)}=\dfrac{2}{3}$.

Chọn C.