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Giải bài 4 trang 90 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Cho hàm số \(f\left( x \right) = \left\{ \begin{array}{l}\frac{{\sqrt {x + 2} - 2}}{{x - 2}}\;khi\;x \ne 2\\\;\;\;\;a\;\;\;\;\;\;\;\;\;\;\;khi\;x = 2\end{array} \right.\). Tìm giá trị của tham số a để hàm số \(y = f\left( x \right)\) liên tục tại \(x = 2\).

♍Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

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Đề bài

Cho hàm số \(f\left( x \right) = \left\{ \begin{array}{l}\frac{{\sqrt {x + 2}  - 2}}{{x - 2}}\;khi\;x \ne 2\\\;\;\;\;a\;\;\;\;\;\;\;\;\;\;\;khi\;x = 2\end{array} \right.\). Tìm giá trị của tham số a để hàm số \(y = f\left( x \right)\) liên tục tại \(x = 2\).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về định nghĩa hàm số liên tục tại một điểm để tìm a: Cho hàm số \(y = f\left( x \right)\) xác định trên khoảng K và \({x_0} \in K\). Hàm số \(y = f\left( x \right)\) được gọi là liên tục tại điểm \({x_0}\) nếu \(\mathop {\lim }\limits_{x \to {x_0}} f\left( x \right) = f\left( {{x_0}} \right)\)

Lời giải chi tiết

Ta có: \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {x + 2}  - 2}}{{x - 2}} = \mathop {\lim }\limits_{x \to 2} \frac{{\left( {\sqrt {x + 2}  - 2} \right)\left( {\sqrt {x + 2}  + 2} \right)}}{{\left( {x - 2} \right)\left( {\sqrt {x + 2}  + 2} \right)}}\) \( = \mathop {\lim }\limits_{x \to 2} \frac{{x - 2}}{{\left( {x - 2} \right)\left( {\sqrt {x + 2}  + 2} \right)}}\)\( = \mathop {\lim }\limits_{x \to 2} \frac{1}{{\sqrt {x + 2}  + 2}} = \frac{1}{{\sqrt {2 + 2}  + 2}} = \frac{1}{4}\) Để hàm số \(y = f\left( x \right)\) liên tục tại \(x = 2\) thì \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = f\left( 2 \right) \Rightarrow a = \frac{1}{4}\)

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