Câu hỏi 15

\(f'\left( -1 \right)=\underset{x\to \left( -1 \right)}{\mathop{\lim }}\,\frac{f\left( x \right)-f\left( -1 \right)}{x+1}\)

Ta có:

\(\begin{array}{l}\mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ + }} \frac{{f\left( x \right) - f\left( { - 1} \right)}}{{x + 1}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ + }} \frac{{\frac{{{x^2} + x + 1}}{x} + 1}}{{x + 1}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ + }} \frac{{{x^2} + 2x + 1}}{{x\left( {x + 1} \right)}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ + }} \frac{{x + 1}}{x} = 0\\\mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ - }} \frac{{f\left( x \right) - f\left( { - 1} \right)}}{{x + 1}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ - }} \frac{{\frac{{{x^2} - x - 1}}{x} + 1}}{{x + 1}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ - }} \frac{{{x^2} - 1}}{{x\left( {x + 1} \right)}} = \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ - }} \frac{{x - 1}}{x} = 2\\ \Rightarrow \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ + }} \frac{{f\left( x \right) - f\left( { - 1} \right)}}{{x + 1}} \ne \mathop {\lim }\limits_{x \to {{\left( { - 1} \right)}^ - }} \frac{{f\left( x \right) - f\left( { - 1} \right)}}{{x + 1}}\end{array}\)

Do đó không tồn tại  \(\underset{x\to \left( -1 \right)}{\mathop{\lim }}\,\frac{f\left( x \right)-f\left( -1 \right)}{x+1}\), vậy không tồn tại đạo hàm của hàm số tại \({{x}_{0}}=-1\).

Chọn D.