Cho hàm số \(f\left( x \right) = \left\{ \begin{array}{l}{x^2} - 3x + 1\;\;khi\;\;x < 0\\\sqrt {{x^2} + 1} \;\;\;\;\;\;khi\;\;x \ge 0\end{array} \right.\). Khi đó:

a) Giới hạn \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = - 1\).

b) Giới hạn \(\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = - 1\).

c) Giới hạn \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = 1\).

d) Giới hạn \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 1\).

a) \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = \mathop {\lim }\limits_{x \to 2} \sqrt {{x^2} + 1}  = \sqrt 5 \).

b) \(\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ - }} \left( {{x^2} - 3x + 1} \right) = 1\).

c) \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ + }} \sqrt {{x^2} + 1}  = 1\).

d) Do \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right)\) nên \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 1\).

Đáp án: a) Sai;  b) Sai;   c) Đúng;   d) Đúng.