Tính giới hạn lim x → + ∞ ( √ x^2 − x − 3 √ x^3 + 1 ) .
Giải thích
Trả lời:−0,5
\(\mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} - x} - \sqrt[3]{{{x^3} + 1}}} \right)\)\( = \mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} - x} - x - \sqrt[3]{{{x^3} + 1}} + x} \right)\)
\[ = \mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} - x} - x} \right) - \mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt[3]{{{x^3} + 1}} - x} \right)\]
\[ = \mathop {\lim }\limits_{x \to + \infty } \frac{{ - x}}{{\sqrt {{x^2} - x} + x}} - \mathop {\lim }\limits_{x \to + \infty } \frac{1}{{\sqrt[3]{{{{\left( {{x^3} + 1} \right)}^2}}} + x\sqrt[3]{{{x^3} + 1}} + {x^2}}}\]
\[ = \mathop {\lim }\limits_{x \to + \infty } \frac{{ - 1}}{{\sqrt {1 - \frac{1}{x}} + 1}} - 0 = - 0,5\].