Problem 18 - Calculus: Early Transcendentals

Question

Prove the statement using the \( \varepsilon \), \( \delta \) definition of a limit and illustrate with a diagram like Figure 9. \( \displaystyle \lim_{x \to -2}(3x + 5) = -1 \)

Step-by-Step Solution

Answer

The limit is confirmed with \( \delta = \frac{\varepsilon}{3} \).

Step 1: Understanding the Objective

We need to prove that the limit of the function \( f(x) = 3x + 5 \) as \( x \to -2 \) is \( -1 \). We will use the \( \varepsilon, \delta \) definition of a limit for this proof.

Step 2: Using the Definition of a Limit

According to the \( \varepsilon, \delta \) definition of a limit, we need to show that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x + 2| < \delta \), it follows that \( |(3x+5) - (-1)| < \varepsilon \).

Step 3: Simplifying the Inequality

Simplify the inequality: \[ |(3x + 5) - (-1)| < \varepsilon. \] This becomes \( |3x + 6| < \varepsilon \), which further simplifies to \( 3|x + 2| < \varepsilon \).

Step 4: Solving for \( \delta \)

From the inequality \( 3|x + 2| < \varepsilon \), solve for \( |x + 2| \) by dividing both sides of the inequality by 3: \[ |x + 2| < \frac{\varepsilon}{3}. \] Choose \( \delta = \frac{\varepsilon}{3} \), which satisfies the condition.

Step 5: Conclusion

For every \( \varepsilon > 0 \), we choose \( \delta = \frac{\varepsilon}{3} \). Then if \( 0 < |x + 2| < \delta \), \( |f(x) + 1| < \varepsilon \) holds true. Thus, \( \lim_{x \to -2}(3x + 5) = -1 \) is proven by the \( \varepsilon, \delta \) definition.

Key Concepts

Limit Proof

In mathematics, proving a limit formally often revolves around the epsilon-delta (\(\varepsilon\), \(\delta\)) definition. This method is essential in calculus to rigorously verify that a function approaches a specific value as the input approaches a given point. To prove the limit of a function such as \(\lim_{x \to -2}(3x + 5) = -1\), you must demonstrate that for any small positive number \(\varepsilon\) (epsilon), there is a corresponding small positive number \(\delta\) (delta). These numbers measure how close \(f(x)\) is to its limit at a certain distance from the given point.

• Begin by expressing the proximity condition: \(|f(x) - L| < \varepsilon\), where \(L\) is the proposed limit.
• Translate this into: \(|(3x + 5) + 1| < \varepsilon\), simplifying to \(|3x + 6| < \varepsilon\).
• The task is then to find \(\delta\) such that whenever \(0 < |x + 2| < \delta\), the inequality \(|3x + 6| < \varepsilon\) holds.

This results in the condition \(3|x + 2| < \varepsilon\), and by solving this, you derive \(\delta = \varepsilon/3\). By choosing such a \(\delta\), the proof satisfies the definition, confirming the limit.

Calculus Concepts

Calculus introduces a variety of concepts which include limits, derivatives, and integrals. Among these, the epsilon-delta definition of a limit stands as a foundational concept that supports much of calculus' theory.Understanding limits through this lens involves transitioning from intuitive notions to precise mathematical formulations. The epsilon-delta approach requires one to:

• Think in terms of closeness: how near must \(f(x)\) be to a limit \(L\) for all \(x\) sufficiently close to \(a\) (provided \(x eq a\))?
• Use inequalities to establish these conditions rigorously.
• Draw connections with real-world applications, such as predicting behavior as a system approaches steady state.

Calculus teaches us that this level of precision is necessary because not all functions behave predictably near every point. This precise framework allows us to handle functions that may have discontinuities, points of non-differentiability, or even behave erratically.

Limits in Mathematics

The concept of limits is integral to many branches of mathematics, particularly calculus. A limit describes the value that a function approaches as the input comes closer to a specified point.When proving limits using the epsilon-delta method, think about it as a challenge to find a relationship between two variables: \(\varepsilon\) (the desired precision around the function's value) and \(\delta\) (the distance to stay within on the input side).

• Limits help in defining derivatives precisely, as they consider the slopes of tangent lines as a function approaches a point.
• In integrals, limits capture the idea of accumulating quantities over continuous regions.
• Functions can behave differently as they approach various points, requiring careful analysis and limits facilitate this understanding.

In real-life applications, limits can predict how a system responds to gradual changes, which is fundamental in fields like engineering, physics, or economics. Grasping these concepts enables one to model and anticipate complex systems effectively.