Cho hàm số \[{\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{\sqrt {{\rm{mx + 1}}} - 1}}{{\rm{x}}}\,\,{\rm{khi}}\,\,{\rm{x}} \ne 0}\\{4{{\rm{x}}^2} + 5{\rm{n}}\,\,{\rm{khi}}\,\,{\rm{x}} = 0}\end{array}} \right.\left( {{\rm{m, n}} \in \mathbb{R}} \right)\] liên tục tại x₀ = 0. Tìm hệ thức liên hệ giữa m và n

Vì\[\mathop {\lim }\limits_{{\rm{x}} \to 2} \left( {{\rm{x}} - 2} \right) = 0 \Rightarrow \]để\[\mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{{\sqrt {3{\rm{x}} + 3} - {\rm{m}}}}{{{\rm{x}} - 2}} = \frac{{\rm{a}}}{{\rm{b}}}\]thì\[\mathop {\lim }\limits_{{\rm{x}} \to 2} \left( {\sqrt {3{\rm{x}} + 3} - {\rm{m}}} \right) = 0\]. Do đó x = 2 là nghiệm của phương trình \[\sqrt {3{\rm{x}} + 3} - {\rm{m}} = 0 \Rightarrow {\rm{m}} = 3\]

Với m = 3 ta được:

\[\mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{{\sqrt {3{\rm{x}} + 3} - 3}}{{{\rm{x}} - 2}} = \mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{{\left( {\sqrt {3{\rm{x}} + 3} - 3} \right)\left( {\sqrt {3{\rm{x}} + 3} + 3} \right)}}{{\left( {{\rm{x}} - 2} \right)\left( {\sqrt {3{\rm{x}} + 3} + 3} \right)}} = \mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{{3{\rm{x}} - 6}}{{\left( {{\rm{x}} - 2} \right)\left( {\sqrt {3{\rm{x}} + 3} + 3} \right)}}\]

\[ = \mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{{3\left( {{\rm{x}} - 2} \right)}}{{\left( {{\rm{x}} - 2} \right)\left( {\sqrt {3{\rm{x}} + 3} + 3} \right)}} = \mathop {\lim }\limits_{{\rm{x}} \to 2} \frac{3}{{\left( {\sqrt {3{\rm{x}} + 3} + 3} \right)}} = \frac{1}{2}\]

\[ \Rightarrow {\rm{a}} = 1,{\rm{b}} = 2 \Rightarrow {\rm{a}} - {\rm{b}} = - 1\]Chọn đáp án C

Đáp án cần chọn là: C

Ta có

\[{\rm{A = }}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to {\rm{1}}} \frac{{{{\rm{x}}^{\rm{n}}} - {\rm{1}}}}{{{{\rm{x}}^{\rm{m}}} - {\rm{1}}}}{\rm{ = }}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to {\rm{1}}} \frac{{{\rm{(x}} - {\rm{1)(}}{{\rm{x}}^{{\rm{n}} - {\rm{1}}}}{\rm{ + }}{{\rm{x}}^{{\rm{n}} - {\rm{2}}}}{\rm{ + }}{{\rm{x}}^{{\rm{n}} - {\rm{3}}}}{\rm{ + }}...{\rm{ + x + 1)}}}}{{{\rm{(x}} - {\rm{1)(}}{{\rm{x}}^{{\rm{m}} - {\rm{1}}}}{\rm{ + }}{{\rm{x}}^{{\rm{m}} - {\rm{2}}}}{\rm{ + }}{{\rm{x}}^{{\rm{m}} - {\rm{3}}}}{\rm{ + }}...{\rm{ + x + 1)}}}}\]

\[ = \mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{{\rm{x}}^{{\rm{n}} - {\rm{1}}}}{\rm{ + }}{{\rm{x}}^{{\rm{n}} - {\rm{2}}}}{\rm{ + }}{{\rm{x}}^{{\rm{n}} - {\rm{3}}}}{\rm{ + }}...{\rm{ + x + 1}}}}{{{{\rm{x}}^{{\rm{m}} - {\rm{1}}}}{\rm{ + }}{{\rm{x}}^{{\rm{m}} - {\rm{2}}}}{\rm{ + }}{{\rm{x}}^{{\rm{m}} - 3}}{\rm{ + }}...{\rm{ + x + 1}}}} = \frac{{1 + 1 + 1 + ... + 1 + 1}}{{1 + 1 + 1 + ... + 1 + 1}} = \frac{{\rm{n}}}{{\rm{m}}}\]

Chọn đáp án D

Đáp án cần chọn là: D