Tính giới hạn \(\mathrm{E}=\lim\limits _{x \rightarrow 7} \frac{\sqrt[3]{4 x-1}-\sqrt{x+2}}{\sqrt[4]{2 x+2}-2}\)
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Lời giải:
Báo sai\(\begin{array}{l} \frac{{\sqrt[3]{{4x - 1}} - \sqrt {x + 2} }}{{\sqrt[4]{{2x + 2}} - 2}} = \frac{{\left( {\sqrt[3]{{4x - 1}} - 3} \right) - \left( {\sqrt {x + 2} - 3} \right)}}{{\sqrt[4]{{2x + 2}} - 2}} = \frac{{\sqrt[3]{{4x - 1}} - 3}}{{\sqrt[4]{{2x + 2}} - 2}} - \frac{{\sqrt {x + 2} - 3}}{{\sqrt[4]{{2x + 2}} - 2}}\\ = \frac{1}{{\sqrt[4]{{2x + 2}} - 2}}\left( {\left( {\sqrt[3]{{4x - 1}} - 3} \right) - \left( {\sqrt {x + 2} - 1} \right)} \right)\\ = \frac{1}{{\sqrt[4]{{2x + 2}} - 2}}\left( {\frac{{4x - 1 - 27}}{{{{\sqrt[3]{{4x - 1}}}^2} + 3\sqrt[3]{{4x - 1}} + 9}} - \frac{{x - 7}}{{\sqrt {x + 2} + 3}}} \right)\\ = \frac{1}{{\sqrt[4]{{2x + 2}} - 2}}\left( {\frac{{4\left( {x - 7} \right)}}{{{{\sqrt[3]{{4x - 1}}}^2} + 3\sqrt[3]{{4x - 1}} + 9}} - \frac{{x - 7}}{{\sqrt {x + 2} + 3}}} \right)\\ = \frac{{x - 7}}{{\sqrt[4]{{2x + 2}} - 2}}\left( {\frac{4}{{{{\sqrt[3]{{4x - 1}}}^2} + 3\sqrt[3]{{4x - 1}} + 9}} - \frac{1}{{\sqrt {x + 2} + 3}}} \right) = {I_1} + {I_2} + {I_3}\\ \mathop {\lim }\limits_{x \to 7} {I_1} = \mathop {\lim }\limits_{x \to 7} \frac{{x - 7}}{{\sqrt[4]{{2x + 2}} - 2}} = \mathop {\lim }\limits_{x \to 7} \frac{{\left( {x - 7} \right)\left( {\sqrt[4]{{2x + 2}} + 2} \right)}}{{\sqrt {2x + 2} - 4}} = \mathop {\lim }\limits_{x \to 7} \frac{{\left( {x - 7} \right)\left( {\sqrt[4]{{2x + 2}} + 2} \right)\left( {\sqrt {2x + 2} + 4} \right)}}{{2x - 14}}\\ = \mathop {\lim }\limits_{x \to 7} \frac{{\left( {x - 7} \right)\left( {\sqrt[4]{{2x + 2}} + 2} \right)\left( {\sqrt {2x + 2} + 4} \right)}}{{2\left( {x - 7} \right)}} = \mathop {\lim }\limits_{x \to 7} \frac{{\left( {\sqrt[4]{{2x + 2}} + 2} \right)\left( {\sqrt {2x + 2} + 4} \right)}}{2} = 16\\ \mathop {\lim }\limits_{x \to 7} {I_2} = \mathop {\lim }\limits_{x \to 7} \frac{4}{{{{\sqrt[3]{{4x - 1}}}^2} + 3\sqrt[3]{{4x - 1}} + 9}} = \frac{4}{{27}}\\ \mathop {\lim }\limits_{x \to 7} {I_3} + \mathop {\lim }\limits_{x \to 7} \frac{1}{{\sqrt {x + 2} + 3}} = \frac{1}{6}\\ E = 16\left( {\frac{4}{{27}} - \frac{1}{6}} \right) = - \frac{8}{{27}} \end{array}\)
