Question

Prove the statement using the\(\varepsilon \), \(\delta \)definition of a limit.

\(\mathop {\lim }\limits_{x \to a} c = c\)

Short answer

It is proved that\(\mathop {\lim }\limits_{x \to a} c = c\).

Step-by-step solution

Describe the given information

It is required to prove\(\mathop {\lim }\limits_{x \to a} c = c\)by using\(\varepsilon \),\(\delta \)definition.

Prove that \(\mathop {\lim }\limits_{x \to a} c = c\)

Consider the limit\(\mathop {\lim }\limits_{x \to a} c = c\).

Let \(\varepsilon \) be a given positive number.

According to the definition with \(\varepsilon \)and\(L = c\), it is required to find a number \(\delta \) such that if \(0 < \left| {x - a} \right| < \delta \) then\(\left| {c - c} \right| < \varepsilon \). But \(\left| {c - c} \right| = 0\), then if \(0 < \left| {x - a} \right| < \delta \) then \(\left| 0 \right| < \varepsilon \)

If \(\delta = \varepsilon \) is chosen then it happens that if \(0 < \left| {x - a} \right| < \delta \)then\(\left( {\left| 0 \right| = \left| {c - c} \right|} \right) < \left( {\delta = \varepsilon } \right)\). Therefore, by the definition of limit\(\mathop {\lim }\limits_{x \to a} c = c\).

Therefore, it is proved that\(\mathop {\lim }\limits_{x \to a} c = c\).