Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Verified step by step guidance

Start by understanding the problem: We need to find the largest open interval around c = 4 where the inequality |f(x) - L| < ε holds, and then determine a δ > 0 such that for all x satisfying 0 < |x - c| < δ, the inequality |f(x) - L| < ε is true.

Substitute the given values into the inequality |f(x) - L| < ε. Here, f(x) = 1/x, L = 1/4, and ε = 0.05. So, the inequality becomes |1/x - 1/4| < 0.05.

To solve |1/x - 1/4| < 0.05, break it into two inequalities: 1/x - 1/4 < 0.05 and 1/x - 1/4 > -0.05. Solve each inequality separately to find the range of x.

For the inequality 1/x - 1/4 < 0.05, rearrange it to find x: 1/x < 0.05 + 1/4. Similarly, for 1/x - 1/4 > -0.05, rearrange it to find x: 1/x > -0.05 + 1/4.

Once you have the range of x from both inequalities, determine the largest open interval around c = 4 where both inequalities are satisfied. Then, choose a δ > 0 such that 0 < |x - 4| < δ ensures |1/x - 1/4| < 0.05.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

In calculus, the limit of a function describes the behavior of the function as its input approaches a certain value. Specifically, for a function f(x) and a point c, we say that the limit of f(x) as x approaches c is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - c| < δ, it follows that |f(x) - L| < ε. This concept is foundational for understanding continuity and differentiability.

Epsilon-Delta Definition

The epsilon-delta definition of a limit formalizes the intuitive idea of limits in calculus. It states that for a function f(x) to approach a limit L as x approaches c, we can make the values of f(x) arbitrarily close to L by choosing x sufficiently close to c. The ε (epsilon) represents how close we want f(x) to be to L, while δ (delta) represents how close x must be to c to achieve that closeness.

Open Interval

An open interval around a point c, denoted as (c - δ, c + δ), consists of all numbers x such that the distance from x to c is less than δ. This concept is crucial in limit analysis, as it defines the range of x values for which the limit condition holds. In the context of the given problem, identifying the largest open interval where the inequality |f(x) - L| < ε is satisfied is essential for determining the behavior of the function near the point c.

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