Q. 12 - Calculus

Question

Write each of the following inequalities in interval notation:

0 < |x − c| < δ

Step-by-Step Solution

Answer

Interval notation is $(c - \delta , c) \cup (c, c + \delta )$

Step 1. Given Information:

Given inequality: 0 < |x − c| < δ 

We want to write given inequalities in interval notation.

Step 2. Solution:

Formal Definition of Limit

The limit

 $$\begin{gathered} \underset{x\to c}{\lim }f \left(x\right) = L means that for all \epsilon > 0, there exists \delta > 0 such that \\ if x \in (c - \delta , c) \cup (c, c + \delta ), then f \left(x\right) \in (L - \epsilon , L + \epsilon ). \end{gathered}$$

Algebraic Definition of Limit

The limit

$$\begin{gathered} \underset{x\to c}{\lim } f \left(x\right) = L means that for all \epsilon > 0, there exists \delta > 0 such that \\ if 0 < |x - c| < \delta , then | f (x) - L| < \epsilon \end{gathered}$$

The two definitions of limit are equivalent we get

$$\begin{gathered} \left(a\right) x \in (c - \delta , c) \cup (c, c + \delta ) if and only if 0 < |x - c| < \delta ; \\ \left(b\right) f \left(x\right) \in (L - \epsilon , L + \epsilon ) if and only if | f (x) - L| < \epsilon . \end{gathered}$$

By using (a) we have;

$0 < |x - c| < \delta \iff x\in x \in (c - \delta , c) \cup (c, c + \delta )$

Therefore the interval notation is $(c - \delta , c) \cup (c, c + \delta )$