Tính \[\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\sqrt[4]{{1 + 4x}} - 1}}{x}\]

Ta có:

\[\begin{array}{*{20}{l}}{\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\sqrt[4]{{1 + 4x}} - 1}\\{ = \sqrt {1 + 2x} - \sqrt {1 + 2x} + \sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}} - \sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}} + \sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\sqrt[4]{{1 + 4x}} - 1}\\{ = \left( {\sqrt {1 + 2x} - 1} \right) + \sqrt {1 + 2x} \left( {\sqrt[3]{{1 + 3x}} - 1} \right) + \sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\left( {\sqrt[4]{{1 + 4x}} - 1} \right)}\end{array}\]

\[\begin{array}{*{20}{l}}{ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\sqrt[4]{{1 + 4x}} - 1}}{x}}\\{ = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\sqrt {1 + 2x} - 1}}{x}} \right) + \mathop {\lim }\limits_{x \to 0} \left( {\sqrt {1 + 2x} .\frac{{\sqrt[3]{{1 + 3x}} - 1}}{x}} \right) + \mathop {\lim }\limits_{x \to 0} \left( {\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\frac{{\sqrt[4]{{1 + 4x}} - 1}}{x}} \right)}\end{array}\]

Tính:

\[\mathop {\lim }\limits_{x \to 0} \left( {\frac{{\sqrt {1 + 2x} - 1}}{x}} \right) = \mathop {\lim }\limits_{x \to 0} \frac{{\left( {\sqrt {1 + 2x} - 1} \right)\left( {\sqrt {1 + 2x} + 1} \right)}}{{x\left( {\sqrt {1 + 2x} + 1} \right)}} = \mathop {\lim }\limits_{x \to 0} \frac{{2x}}{{x\left( {\sqrt {1 + 2x} + 1} \right)}} = \mathop {\lim }\limits_{x \to 0} \frac{2}{{\sqrt {1 + 2x} + 1}} = \frac{2}{{1 + 1}} = 1\]\[\begin{array}{l}\mathop {lim}\limits_{x \to 0} \left( {\sqrt {1 + 2x} .\frac{{\sqrt[3]{{1 + 3x}} - 1}}{x}} \right)\\ = \mathop {lim}\limits_{x \to 0} \left( {\sqrt {1 + 2x} .\frac{{\left( {\sqrt[3]{{1 + 3x}} - 1} \right)\left[ {{{\left( {\sqrt[3]{{1 + 3x}}} \right)}^2} + \sqrt[3]{{1 + 3x}} + 1} \right]}}{{x.\left[ {{{\left( {\sqrt[3]{{1 + 3x}}} \right)}^2} + \sqrt[3]{{1 + 3x}} + 1} \right]}}} \right)\\ = \mathop {lim}\limits_{x \to 0} \left( {\sqrt {1 + 2x} .\frac{{3x}}{{x.\left[ {{{\left( {\sqrt[3]{{1 + 3x}}} \right)}^2} + \sqrt[3]{{1 + 3x}} + 1} \right]}}} \right)\\ = \mathop {lim}\limits_{x \to 0} \left( {\frac{{3\sqrt {1 + 2x} }}{{\left[ {{{\left( {\sqrt[3]{{1 + 3x}}} \right)}^2} + \sqrt[3]{{1 + 3x}} + 1} \right]}}} \right) = \frac{{3.1}}{{1 + 1 + 1}} = 3\end{array}\]

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

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

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

\[ = \mathop {lim}\limits_{x \to 0} \frac{{4\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}}}{{\left[ {{{\left( {\sqrt[4]{{1 + 4x}}} \right)}^3} + {{\left( {\sqrt[4]{{1 + 4x}}} \right)}^2} + \sqrt[4]{{1 + 4x}} + 1} \right]}} = \frac{{4.1.1}}{{1 + 1 + 1 + 1}} = 1\]

Vậy\[\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + 2x} .\sqrt[3]{{1 + 3x}}.\sqrt[4]{{1 + 4x}} - 1}}{x} = 1 + 1 + 1 = 3\]