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Lý thuyết Đường tiệm cận của đồ thị hàm số Toán 12 Cánh Diều

1. Đường tiệm cận ngang

🍰Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Cánh diều

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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1. Đường tiệm cận ngang

Đường thẳng \(y = {y_0}\) gọi là đường tiệm cận ngang (hay tiệm cận ngang) của đồ thị hàm số y = f(x) nếu \(\mathop {\lim }\limits_{x \to  + \infty } f(x) = {y_0}\) hoặc \(\mathop {\lim }\limits_{x \to  - \infty } f(x) = {y_0}\)

Ví dụ:🦋 Tìm TCN của đồ thị hàm số \(y = f(x) = \frac{{3x - 2}}{{x + 1}}\)

Ta có: \(\mathop {\lim }\limits_{x \to  + \infty } \frac{{3x - 2}}{{x + 1}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{3x - 2}}{{x + 1}} = 3\)Vậy đồ thị hàm số f(x) có TCN là y = 3.

2. Đường tiệm cận đứng

Đường thẳng \(x = {x_0}\) gọi là đường tiệm cận đứng (gọi tắt là tiệm cận đứng) của đồ thị hàm số y = f(x) nếu ít nhất một trong các điều kiện sau được thỏa mãn:\(\mathop {\lim }\limits_{x \to {x_0}^ + } f(x) =  + \infty ;\mathop {\lim }\limits_{x \to {x_0}^ + } f(x) =  - \infty ;\mathop {\lim }\limits_{x \to {x_0}^ - } f(x) =  + \infty ;\mathop {\lim }\limits_{x \to {x_0}^ - } f(x) =  - \infty \);

Ví dụ: ♍Tìm TCĐ của đồ thị hàm số \(y = f(x) = \frac{{3 - x}}{{x + 2}}\)

Ta có: \(\mathop {\lim }\limits_{x \to  - {2^ + }} \frac{{3x - 2}}{{x + 2}} =  + \infty \)Vậy đồ thị hàm số có TCĐ là x = -2

3.Đường tiệm cận xiên

Đường thẳng \(y = ax + b(a \ne 0)\) gọi là đường tiệm cận xiên (gọi tắt là tiệm cận xiên) của đồ thị hàm số y = f(x) nếu \(\mathop {\lim }\limits_{x \to  + \infty } f(x) = \left[ {f(x) - (ax + b)} \right] = 0\) hoặc \(\mathop {\lim }\limits_{x \to  - \infty } f(x) = \left[ {f(x) - (ax + b)} \right] = 0\)

Ví dụ: 🐭Tìm TCX của đồ thị hàm số \(y = f(x) = x + \frac{1}{{x + 2}}\)

Ta có: \(\mathop {\lim }\limits_{x \to  + \infty } \left[ {f(x) - x} \right] = \mathop {\lim }\limits_{x \to  + \infty } \frac{1}{{x + 2}} = 0\)Vậy đồ thị hàm số có TCX là y = x

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