Tìm giá trị của tham số \(m\) để hàm số sau liên tục tại \(x=0\).
\(f(x)=\left\{\begin{array}{l}\frac{\sqrt{1-x}-\sqrt{1+x}}{x}, x<0 m+\frac{1-x}{1+x}, x \geq 0\end{array}\right.\).
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Ta có:
\(\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\left(m+\frac{1-x}{1+x}\right)=m+1\)
\(\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left(\frac{\sqrt{1-x}-\sqrt{1+x}}{x}\right)=\lim _{x \rightarrow 0^{-}}\left(\frac{-2 x}{x(\sqrt{1-x}+\sqrt{1+x})}\right)=\lim _{x \rightarrow 0^{-}}\left(\frac{-2}{(\sqrt{1-x}+\sqrt{1+x})}\right)=-1\)
\(f(0)=m+1\)
\(f(x)\) liên tục tại \(x=0\) khi và chỉ khi
\(\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}} f(x)=f(0) \Leftrightarrow m+1=-1 \Leftrightarrow m=-2\).
Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 2 mang đến cho thí sinh một trải nghiệm thi cử mới mẻ, bám sát chương trình GDPT 2018, kiểm tra khả năng Giải Quyết Vấn Đề, Tư Duy Sáng Tạo, Giao Tiếp và Hợp Tác. Với thời lượng 195 phút, bài thi bao gồm ba phần: Toán Học Và Xử Lí Số Liệu/Tư Duy Định Lượng, Văn Học - Ngôn Ngữ/Tư Duy Định Tính và Khoa Học/Tiếng Anh. Mỗi phần đều có dạng thức câu hỏi phong phú như trắc nghiệm nhiều phương án lựa chọn và điền đáp án, đảm bảo đánh giá toàn diện năng lực học sinh trên nhiều khía cạnh học thuật khác nhau.
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