Cho hàm số \(y = f\left( x \right) = \frac{{2\sqrt {1 + x}  - \sqrt[3]{{8 - x}}}}{x}.

* Hướng dẫn giải

\(\mathop {\lim }\limits_{x \to 0} {\mkern 1mu} {\mkern 1mu} f\left( x \right) = \mathop {\lim }\limits_{x \to 0} {\mkern 1mu} \frac{{2\sqrt {1 + x}  - 2 + 2 - \sqrt[3]{{8 - x}}}}{x} = \mathop {\lim }\limits_{x \to 0} {\mkern 1mu} \frac{{2\left[ {\frac{{\left( {1 + x} \right) - 1}}{{\sqrt {1 + x}  + 1}}} \right] + \frac{{8 - \left( {8 - x} \right)}}{{4 + 2\sqrt[3]{{8 - x}} + \sqrt[3]{{{{\left( {8 - x} \right)}^2}}}}}}}{x}a\)

\( = \mathop {\lim }\limits_{x \to 0} {\mkern 1mu} \left( {\frac{2}{{\sqrt {1 + x}  + 1}} + \frac{1}{{4 + 2\sqrt[3]{{8 - x}} + \sqrt[3]{{{{\left( {8 - x} \right)}^2}}}}}} \right) = \frac{{13}}{{12}}\)