Problem 11 - Calculus with Concepts in Calculus

Question

Use the definition of limit to verify the given limit. $$ \lim _{x \rightarrow 0}(4 x+7)=7 $$

Step-by-Step Solution

Answer

The limit is verified with \( \delta = \epsilon / 4 \).

Step 1: Understanding the Limit Definition

The definition of the limit states that for every number \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \(|f(x) - L| < \epsilon\). In this case, \( f(x) = 4x + 7 \), \( L = 7 \), and \( a = 0 \). We want to show that \( \lim_{x \to 0}(4x + 7) = 7 \) using this definition.

Step 2: Set Up the Inequality

We need \(|f(x) - L| = |(4x + 7) - 7| = |4x|\) to be less than \( \epsilon \). So, we need \(|4x| < \epsilon\).

Step 3: Solve for x

To satisfy the inequality \(|4x| < \epsilon\), we simplify to get \(|x| < \epsilon / 4\).

Step 4: Determine Delta

From the inequality \(|x| < \epsilon / 4\), we can choose \( \delta = \epsilon / 4 \). This choice ensures that whenever \( 0 < |x| < \delta \), \(|4x| < \epsilon\), demonstrating the limit according to the definition.

Key Concepts

Understanding the Epsilon-Delta Definition

The epsilon-delta definition of a limit is a formal way to express the concept of a limit in calculus. It might seem complex at first, but it's quite straightforward when you break it down. The basic idea is to show that as a variable, say \(x\), gets arbitrarily close to a certain point \(a\), the function \(f(x)\) approaches a specific value \(L\).

To state this formally, for every tiny distance \(\epsilon > 0\) from \(L\) that we can think of, we can find another tiny distance \(\delta > 0\) such that whenever \(x\) is within \(\delta\) distance from \(a\) (but not exactly at \(a\)), \(f(x)\) is within \(\epsilon\) distance from \(L\).

• \(|f(x) - L| < \epsilon\) means that the distance from \(f(x)\) to \(L\) is less than \(\epsilon\).
• \(0 < |x - a| < \delta\) restricts \(x\) to be close to \(a\), but not exactly at \(a\).

In applying this to the problem \(\lim_{x \to 0}(4x + 7) = 7\), we set \(a = 0\) and \(L = 7\). The goal is to find a suitable \(\delta\) for every \(\epsilon\) that makes \(|4x| < \epsilon\) hold true. This process helps solidify the concept of limits in calculus by ensuring exactness in how close \(x\) needs to be to \(a\) to keep \(f(x)\) close to \(L\).

Interpreting Limits of Functions

In calculus, understanding the limits of functions is essential. Limits allow us to analyze the behavior of functions as they approach a specific input value. Essentially, a limit tells us what value a function is approaching as its input approaches a certain point.

For the function \(f(x) = 4x + 7\), we're examining the limit as \(x\) approaches 0. This means we're investigating what value \(4x + 7\) gets closer to when \(x\) becomes very small, close to 0 without actually being zero. Here, we are aiming to show that as \(x\) gets nearer to 0, \(4x + 7\) approaches 7. Hence, we conclude that \(\lim_{x \to 0}(4x + 7) = 7\).

• Think of limits as a way to "predict" where a function is heading.
• It provides foundational understanding for derivatives and integrals later in calculus.

The consistency of limits across all scales means they help verify when functions can be smoothly understood as being near certain values even when direct substitution doesn’t give a clear result.

Applying Calculus Problem Solving Techniques

Problem solving in calculus often involves a structured, methodical approach to find a solution, especially when using the epsilon-delta definition. Here, we start by recognizing the function, the point we're approaching, and the limit we're testing. With \(f(x) = 4x + 7\), \(a = 0\), and \(L = 7\), our task becomes clear: use these values effectively to find the necessary \(\delta\).

The goal is to demonstrate how \(|4x| < \epsilon\) implies that \(|x| < \epsilon / 4\) and thereby choose \(\delta = \epsilon / 4\).

• Identify the terms \(\epsilon\), \(\delta\), and align them with the function's structure.
• Solve inequalities that drive the proof and show limit validation.

By setting up an inequality and solving for \(x\), we've successfully demonstrated the limit using the epsilon-delta definition. This step-by-step analysis not only advances our understanding of limits but also hones our technique in calculus problem solving. Each part of the process builds confidence in tackling more complex calculus problems in future chapters.