Tính giới hạn \(\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt[3]{{1 + 4x}} - 1}}{x}.\)

* Hướng dẫn giải

\(\mathop {{\rm{lim}}}\limits_{x \to 0} \left( {\frac{{\sqrt[3]{{1 + 4x}} - 1}}{{\rm{x}}}{\rm{\;}}} \right) = \mathop {\lim }\limits_{x \to 0} \frac{{4x}}{{x\left( {\sqrt[3]{{{{\left( {1 + 4x} \right)}^2}}} + \sqrt[3]{{1 + 4x}} + 1} \right)}} = \mathop {\lim }\limits_{x \to 0} \frac{4}{{\left( {\sqrt[3]{{{{\left( {1 + 4x} \right)}^2}}} + \sqrt[3]{{1 + 4x}} + 1} \right)}} = \frac{4}{3}\)