Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Verified step by step guidance

1. Step 1: Understand the problem statement. We are given that |f(x) - 5| < 0.1 for 0 < x < 5. We need to find a δ > 0 such that |f(x) - 5| < 0.1 whenever 0 < |x - 2| < δ.
2. Step 2: Recognize that this is a problem about continuity and limits. Specifically, it is related to the definition of a limit at a point, where we want to ensure that f(x) is close to 5 when x is close to 2.
3. Step 3: Consider the interval 0 < x < 5. Since we want 0 < |x - 2| < δ, we are focusing on values of x that are close to 2 but still within the interval (0, 5).
4. Step 4: Choose δ such that the interval (2 - δ, 2 + δ) is contained within (0, 5). This ensures that for any x in (2 - δ, 2 + δ), the condition 0 < x < 5 is satisfied, and thus |f(x) - 5| < 0.1 holds.
5. Step 5: Determine the largest possible δ by considering the boundaries of the interval (0, 5) and the point x = 2. The largest δ will be the minimum of the distances from 2 to the endpoints of the interval (0, 5).

Key Concepts

Epsilon-Delta Definition of Limit

The epsilon-delta definition of a limit formalizes the concept of limits in calculus. It states that for every ε (epsilon) greater than 0, there exists a δ (delta) such that if the distance between x and a point a is less than δ, then the distance between f(x) and L (the limit) is less than ε. This definition is crucial for understanding continuity and limits in calculus.

Absolute Value Inequalities

Absolute value inequalities express the distance between two values. The inequality |f(x) - 5| < 0.1 indicates that the function f(x) is within 0.1 units of the value 5. Understanding how to manipulate and interpret these inequalities is essential for solving problems related to limits and continuity.

Neighborhoods in Calculus

In calculus, a neighborhood around a point a is defined as the set of all points within a certain distance from a. Specifically, the notation |x - 2| < δ describes a neighborhood around the point x = 2. This concept is important for establishing conditions under which a function behaves predictably, particularly in the context of limits and continuity.