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Giải bài 5.36 trang 88 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Giới hạn \(\mathop {\lim }\limits_{x \to - \infty } \frac{{\sqrt {{x^2} + 2} - x}}{{|x|}}\) là

𝓡Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Giới hạn \(\mathop {\lim }\limits_{x \to  - \infty } \frac{{\sqrt {{x^2} + 2}  - x}}{{x}}\) là

A. \( + \infty \)              

B. 0                     

C. - 2                   

D. Không tồn tại.

Phương pháp giải - Xem chi tiết

- ℱCác quy tắc tính giới hạn hữu hạn tại một điểm cũng đúng cho giới hạn hữu hạn tại vô cực.

- Với c là hằng số, ta có: \(\mathop {\lim }\limits_{x \to  + \infty } c = c,\mathop {\lim }\limits_{x \to  - \infty } c = c\) - Với k là một số nguyên dương, ta có: \(\mathop {\lim }\limits_{x \to  + \infty } \frac{1}{{{x^k}}} = 0,\mathop {\lim }\limits_{x \to  - \infty } \frac{1}{{{x^k}}} = 0\). Đối với bài tập trên, ta có thể nhóm hạng tử số mũ cao nhất ra ngoài rồi rút gọn.

Lời giải chi tiết

Đáp án C\(\mathop {\lim }\limits_{x \to  - \infty } \frac{{\sqrt {{x^2} + 2}  - x}}{{x}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{|x|\sqrt {1 + \frac{2}{{{x^2}}}}  - x}}{{x}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{ - x\sqrt {1 + \frac{2}{{{x^2}}}}  - x}}{{ x}} = \mathop {\lim }\limits_{x \to  - \infty } \left( {-\sqrt {1 + \frac{2}{{{x^2}}}}  - 1} \right) =- 2\)

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Tham Gia Group Dành Cho Lớp 11 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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