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Giải bài 24 trang 76 sách bài tập toán 11 - Cánh diều

Cho số thực \(a\) và hàm số \(f\left( x \right)\) thoả mãn \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = - \infty \).

🃏Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

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Đề bài

Cho số thực \(a\) và hàm số \(f\left( x \right)\) thoả mãn \(\mathop {\lim }\limits_{x \to a} f\left( x \right) =  - \infty \). Chứng minh rằng \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - 3}}{{2f\left( x \right) + 1}} = \frac{1}{2}\).

Phương pháp giải - Xem chi tiết

Chia cả tử và mẫu của biểu thức \(\frac{{f\left( x \right) - 3}}{{2f\left( x \right) + 1}}\) cho \(f\left( x \right)\), rồi sử dụng các định lí về giới hạn hàm số.

Lời giải chi tiết

Ta có: \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - 3}}{{2f\left( x \right) + 1}} = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)\left[ {1 - \frac{3}{{f\left( x \right)}}} \right]}}{{f\left( x \right)\left[ {2 + \frac{1}{{f\left( x \right)}}} \right]}} = \mathop {\lim }\limits_{x \to a} \frac{{1 - \frac{3}{{f\left( x \right)}}}}{{2 + \frac{1}{{f\left( x \right)}}}} = \frac{{\mathop {\lim }\limits_{x \to a} 1 - \mathop {\lim }\limits_{x \to a} \frac{3}{{f\left( x \right)}}}}{{\mathop {\lim }\limits_{x \to a} 2 + \mathop {\lim }\limits_{x \to a} \frac{1}{{f\left( x \right)}}}}\)\( = \frac{{1 - 0}}{{2 + 0}} = \frac{1}{2}\).

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