Chapter 1: Q. 12 (page 107)

Write each of the following inequalities in interval notation:
0 < |x − c| < δ

Short Answer

Expert verified

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

Step by step solution

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

$\lim_{x \to c} f (x) = L m e a n s t h a t f o r a l l ε > 0, t h e r e e x i s t s δ > 0 s u c h t h a t i f x \in (c - δ, c) \cup (c, c + δ), t h e n f (x) \in (L - ε, L + ε) .$

Algebraic Definition of Limit

The limit

\lim_{x \to c} f (x) = L m e a n s t h a t f o r a l l ε > 0, t h e r e e x i s t s δ > 0 s u c h t h a t i f 0 < | x - c | < δ, t h e n | f (x) - L | < ε

The two definitions of limit are equivalent we get

$(a) x \in (c - δ, c) \cup (c, c + δ) i f a n d o n l y i f 0 < | x - c | < δ; (b) f (x) \in (L - ε, L + ε) i f a n d o n l y i f | f (x) - L | < ε .$

By using (a) we have;

$0 < | x - c | < δ \Leftrightarrow x \in x \in (c - δ, c) \cup (c, c + δ)$

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

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Write each of the following inequalities in interval notation:
0 < |x − c| < δ

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Write each of the following inequalities in interval notation:0 < |x − c| < δ

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Geometry

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Write each of the following inequalities in interval notation:
0 < |x − c| < δ