Cho hàm số f ( x ) = ⎧ ⎨ ⎩ 3 − √ 4 − x 4 k h i x ≠ 0 1 4 k h i x = 0 . Tính f′(0).

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Cho hàm số \[f(x) = \left\{ {\begin{array}{*{20}{c}}{\frac{{3 - \sqrt {4 - x} }}{4}\,\,khi\,\,x \ne 0}\\{\frac{1}{4}\,\,khi\,\,x = 0}\end{array}} \right.\]. Tính f′(0).

\[{\rm{f'}}\left( {\rm{0}} \right){\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{.}}\]

\[{\rm{f'}}\left( {\rm{0}} \right){\rm{ = }}\frac{{\rm{1}}}{{{\rm{16}}}}{\rm{.}}\]

\[{\rm{f'}}\left( {\rm{0}} \right){\rm{ = }}\frac{{\rm{1}}}{{{\rm{32}}}}{\rm{.}}\]

Không tồn tại

Giải thích

Xét \[\mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{{\rm{f}}\left( {\rm{x}} \right) - {\rm{f}}\left( {\rm{0}} \right)}}{{{\rm{x}} - 0}} = \mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{\frac{{3 - \sqrt {4 - {\rm{x}}} }}{4} - \frac{1}{4}}}{{\rm{x}}} = \mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{2 - \sqrt {4 - {\rm{x}}} }}{{4{\rm{x}}}}\]

\[ = \mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{\left( {2 - \sqrt {4 - {\rm{x}}} } \right)\left( {2 + \sqrt {4 - {\rm{x}}} } \right)}}{{4{\rm{x}}\left( {2 + \sqrt {4 - {\rm{x}}} } \right)}} = \mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{\rm{x}}}{{4{\rm{x}}\left( {2 + \sqrt {4 - {\rm{x}}} } \right)}} = \mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{1}{{4\left( {2 + \sqrt {4 - {\rm{x}}} } \right)}} = \frac{1}{{16}}.\]

Đáp án cần chọn là: B