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Giải bài 5.17 trang 83 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hàm số \(g\left( x \right) = \sqrt {{x^2} + 2x} - \sqrt {{x^2} - 1} - 2m\) với m là tham số

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

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Đề bài

Cho hàm số \(g\left( x \right) = \sqrt {{x^2} + 2x}  - \sqrt {{x^2} - 1}  - 2m\) với m là tham số. Biết \(\mathop {\lim }\limits_{x \to  + \infty } g\left( x \right) = 0\), tìm giá trị của m.

Phương pháp giải - Xem chi tiết

- Các quy tắc tính giới hạn hữu hạn tại một điểm c🦹ũng đúng cho giới h✃ạn 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: \(\mathop {\lim }\limits_{x \to  + \infty } \frac{1}{{{x^k}}} = 0,\mathop {\lim }\limits_{x \to  - \infty } \frac{1}{{{x^k}}} = 0\)

Lời giải chi tiết

Ta có: \(g\left( x \right) = \frac{{2x + 1}}{{\sqrt {{x^2} + 2x}  + \sqrt {{x^2} - 1} }} - 2m = \frac{{2 + \frac{1}{x}}}{{\sqrt {1 + \frac{2}{x}}  + \sqrt {1 - \frac{1}{{{x^2}}}} }} - 2m\)Do đó, \(\mathop {\lim }\limits_{x \to  + \infty } g\left( x \right) = \mathop {\lim }\limits_{x \to  + \infty } \frac{{2 + \frac{1}{x}}}{{\sqrt {1 + \frac{2}{x}}  + \sqrt {1 - \frac{1}{{{x^2}}}} }} - 2m = 1 - 2m\)Để \(\mathop {\lim }\limits_{x \to  + \infty } g\left( x \right) = 0\) thì \(1 - 2m = 0 \Leftrightarrow m = \frac{1}{2}\)

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