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Giải bài tập 1.17 trang 25 SGK Toán 12 tập 1 - Kết nối tri thức

Đường thẳng \(x = 1\) có phải là tiệm cận đứng của đồ thị hàm số \(y = \frac{{{x^2} + 2x - 3}}{{x - 1}}\) không?

🐲Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Đường thẳng \(x = 1\) có phải là tiệm cận đứng của đồ thị hàm số \(y = \frac{{{x^2} + 2x - 3}}{{x - 1}}\) không?

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về khái niệm tiệm cận đứng của đồ thị hàm số để tìm 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 \(y = f\left( x \right)\) nếu ít nhất một trong các điều kiện sau được thỏa mãn: \(\mathop {\lim }\limits_{x \to x_0^ + } f\left( x \right) =  + \infty \); \(\mathop {\lim }\limits_{x \to x_0^ - } f\left( x \right) =  - \infty \); \(\mathop {\lim }\limits_{x \to x_0^ + } f\left( x \right) =  - \infty \); \(\mathop {\lim }\limits_{x \to x_0^ - } f\left( x \right) =  + \infty \)

Lời giải chi tiết

Ta có: \(\mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {1^ + }} \frac{{{x^2} + 2x - 3}}{{x - 1}} = \mathop {\lim }\limits_{x \to {1^ + }} \frac{{\left( {x - 1} \right)\left( {x + 3} \right)}}{{x - 1}} = \mathop {\lim }\limits_{x \to {1^ + }} \left( {x + 3} \right) = 4\) \(\mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {1^ - }} \frac{{{x^2} + 2x - 3}}{{x - 1}} = \mathop {\lim }\limits_{x \to {1^ - }} \frac{{\left( {x - 1} \right)\left( {x + 3} \right)}}{{x - 1}} = \mathop {\lim }\limits_{x \to {1^ - }} \left( {x + 3} \right) = 4\) Do đó, đường thẳng \(x = 1\) không là tiệm cận đứng của đồ thị hàm số \(y = \frac{{{x^2} + 2x - 3}}{{x - 1}}\).

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