Bài 27. Tìm các giới hạn sau (nếu có) :
a. \(\mathop {\lim }\limits_{x \to {2^ + }} {{\left| {x – 2} \right|} \over {x – 2}}\)
b. \(\mathop {\lim }\limits_{x \to {2^ – }} {{\left| {x – 2} \right|} \over {x – 2}}\)
c. \(\mathop {\lim }\limits_{x \to 2} {{\left| {x – 2} \right|} \over {x – 2}}\)
a. Với mọi \(x > 2\), ta có \(\left| {x – 2} \right| = x – 2.\) Do đó :
\(\mathop {\lim }\limits_{x \to {2^ + }} {{\left| {x – 2} \right|} \over {x – 2}} = \mathop {\lim }\limits_{x \to {2^ + }} {{x – 2} \over {x – 2}} = \mathop {\lim }\limits_{x \to {2^ + }} 1 = 1\)
b. Với mọi \(x < 2\), ta có \(|x – 2| = 2 – x\). Do đó :
\(\mathop {\lim }\limits_{x \to {2^ – }} {{\left| {x – 2} \right|} \over {x – 2}} = \mathop {\lim }\limits_{x \to {2^ – }} {{2 – x} \over {x – 2}} = \mathop {\lim }\limits_{x \to {2^ – }} – 1 = – 1\)
c. Vì \(\mathop {\lim }\limits_{x \to {2^ + }} {{\left| {x – 2} \right|} \over {x – 2}} \ne \mathop {\lim }\limits_{x \to {2^ – }} {{\left| {x – 2} \right|} \over {x – 2}}\) nên không tồn tại \(\mathop {\lim }\limits_{x \to 2} {{\left| {x – 2} \right|} \over {x – 2}}\)