lim[căn(4x^2 -3x +4) +3x]/[căn(x^2 + x + 1) -1] (x —> âm vô cực) 04/07/2021 Bởi Arianna lim[căn(4x^2 -3x +4) +3x]/[căn(x^2 + x + 1) -1] (x —> âm vô cực)
Đáp án: \[\mathop {\lim }\limits_{x \to – \infty } \frac{{\sqrt {4{x^2} – 3x + 4} + 3x}}{{\sqrt {{x^2} + x + 1} – 1}} = – 1\] Giải thích các bước giải: Ta có: \(\begin{array}{l}\mathop {\lim }\limits_{x \to – \infty } \frac{{\sqrt {4{x^2} – 3x + 4} + 3x}}{{\sqrt {{x^2} + x + 1} – 1}}\\ = \mathop {\lim }\limits_{x \to – \infty } \frac{{\left| x \right|\sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3x}}{{\left| x \right|.\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – 1}}\\ = \mathop {\lim }\limits_{x \to – \infty } \frac{{ – x\sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3x}}{{ – x.\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – 1}}\,\,\,\,\,\,\left( {x \to – \infty \Rightarrow x < 0 \Rightarrow \left| x \right| = – x} \right)\\ = \mathop {\lim }\limits_{x \to – \infty } \frac{{ – \sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3}}{{ – \sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – \frac{1}{x}}}\\ = \frac{{ – \sqrt 4 + 3}}{{ – \sqrt 1 }} = \frac{1}{{ – 1}} = – 1\end{array}\) Bình luận
Đáp án:
\[\mathop {\lim }\limits_{x \to – \infty } \frac{{\sqrt {4{x^2} – 3x + 4} + 3x}}{{\sqrt {{x^2} + x + 1} – 1}} = – 1\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\mathop {\lim }\limits_{x \to – \infty } \frac{{\sqrt {4{x^2} – 3x + 4} + 3x}}{{\sqrt {{x^2} + x + 1} – 1}}\\
= \mathop {\lim }\limits_{x \to – \infty } \frac{{\left| x \right|\sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3x}}{{\left| x \right|.\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – 1}}\\
= \mathop {\lim }\limits_{x \to – \infty } \frac{{ – x\sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3x}}{{ – x.\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – 1}}\,\,\,\,\,\,\left( {x \to – \infty \Rightarrow x < 0 \Rightarrow \left| x \right| = – x} \right)\\
= \mathop {\lim }\limits_{x \to – \infty } \frac{{ – \sqrt {4 – \frac{3}{x} + \frac{4}{{{x^2}}}} + 3}}{{ – \sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}} – \frac{1}{x}}}\\
= \frac{{ – \sqrt 4 + 3}}{{ – \sqrt 1 }} = \frac{1}{{ – 1}} = – 1
\end{array}\)