Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L


d. If |x−a|<δ, then a−δ<x<a+δ.

Verified step by step guidance

1. Step 1: Understand the statement: The statement is about the definition of the limit of a function as x approaches a. It involves the concept of a delta (δ) neighborhood around the point a.
2. Step 2: Recall the definition of a limit: The limit of f(x) as x approaches a is L, written as \( \lim_{{x \to a}} f(x) = L \), if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
3. Step 3: Analyze the given condition: The condition \( |x - a| < \delta \) implies that x is within a distance δ from a. This means x is in the interval (a - δ, a + δ).
4. Step 4: Verify the statement: The statement 'If \( |x - a| < \delta \), then \( a - \delta < x < a + \delta \)' is indeed true by the definition of absolute value and the concept of a δ-neighborhood.
5. Step 5: Conclusion: The statement is true because it directly follows from the definition of the absolute value and the concept of a δ-neighborhood around a point a.

Key Concepts

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. In this case, lim x→a f(x) = L means that as x gets closer to a, the function f(x) approaches the value L. Understanding limits is crucial for analyzing continuity and the behavior of functions near specific points.

Delta-Epsilon Definition of a Limit

The delta-epsilon definition formalizes the concept of limits. It states that for every ε (epsilon) greater than 0, there exists a δ (delta) such that if |x - a| < δ, then |f(x) - L| < ε. This definition is essential for proving the validity of limits and understanding the precise conditions under which a function approaches a limit.

Inequalities and Intervals

Inequalities are used to describe the range of values that a variable can take. The statement |x - a| < δ implies that x lies within the interval (a - δ, a + δ). Understanding how to manipulate and interpret these inequalities is vital for analyzing the behavior of functions in the context of limits and continuity.