Problem 19 - Calculus Early Transcendentals

Question

Prove the statement using the \(\varepsilon, \delta\) definition of limit. \(\lim _{x \rightarrow 3} \frac{x}{5}=\frac{3}{5}\)

Step-by-Step Solution

Answer

Limit is proved by showing \(\delta=5\varepsilon\) works for every \(\varepsilon>0\).

Step 1: Understand the Definition

To prove that \(\lim _{x \rightarrow 3} \frac{x}{5}=\frac{3}{5}\) using the \(\varepsilon, \delta\) definition, we need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(|x-3|<\delta\) then \(\left|\frac{x}{5}-\frac{3}{5}\right|<\varepsilon\).

Step 2: Simplify the Expression

We start with the expression \(\left|\frac{x}{5} - \frac{3}{5}\right|\). Simplifying this gives \(\frac{|x - 3|}{5}\). Thus, we need to ensure \(\frac{|x - 3|}{5} < \varepsilon\).

Step 3: Solve for δ in Terms of ε

To solve \(\frac{|x-3|}{5} < \varepsilon\), multiply both sides by 5 to obtain \(|x - 3| < 5\varepsilon\). Set \(\delta = 5\varepsilon\).

Step 4: Choose an Appropriate δ

Since we want \(|x - 3| < \delta\) to imply \(\left|\frac{x}{5} - \frac{3}{5}\right| < \varepsilon\), we choose \(\delta = 5\varepsilon\). This choice guarantees that any \(|x - 3| < 5\varepsilon\) will satisfy the initial condition \(\left|\frac{x}{5} - \frac{3}{5}\right| < \varepsilon\).

Step 5: Conclude the Proof

For every \(\varepsilon > 0\), we chose \(\delta = 5\varepsilon\). If \(|x-3|<\delta\), then \(|x-3|<5\varepsilon\), which ensures \(\left|\frac{x}{5} - \frac{3}{5}\right|<\varepsilon\). Therefore, by the \(\varepsilon, \delta\) definition of a limit, \(\lim _{x \rightarrow 3} \frac{x}{5}=\frac{3}{5}\).

Key Concepts

Calculus Limit Proof

Understanding how to prove limits is fundamental in calculus. In this exercise, we're given the task to prove that the limit of \( \frac{x}{5} \) as \( x \) approaches 3 is \( \frac{3}{5} \). The first step in a calculus limit proof involves carefully applying the definition of a limit. This entails using the epsilon-delta \( (\varepsilon, \delta) \) method, a rigorous approach to validate limit properties.

Begin by interpreting the limit statement: \( \lim_{x \to 3} \frac{x}{5} = \frac{3}{5} \). According to the epsilon-delta definition, for every positive \( \varepsilon \) (no matter how small), you must find a positive \( \delta \) such that when \(|x - 3| < \delta\), it implies \( \left| \frac{x}{5} - \frac{3}{5} \right| < \varepsilon \).

The proof consists of several steps:

• Breaking down and simplifying the expression that's supposed to be less than \( \varepsilon \).
• Solving for \( \delta \) in terms of \( \varepsilon \).
• Showing the steps through logical math operations and concluding the proof with the chosen \( \delta \).

This structured approach ensures that you're comprehensive and methodical, aligning with mathematical precision required in calculus.

Epsilon-Delta Limit

The epsilon-delta limit is an essential concept in calculus, providing a precise mathematical way to define what limits mean. This approach is popular in calculus because it sets the groundwork for a solid understanding of continuous functions and derivatives.

Consider the equation derived from the limit statement: \( \lim_{x \rightarrow 3} \frac{x}{5} = \frac{3}{5} \). Here, we seek \( \delta > 0 \) based on a given \( \varepsilon > 0 \). The principle is:

1. For every \( \varepsilon > 0 \), there is \( \delta > 0 \).
2. If \( |x - 3| < \delta \), then \( \left| \frac{x}{5} - \frac{3}{5} \right| < \varepsilon \).

In simpler terms, the epsilon-delta definition doesn't just say that approaching \( x = 3 \) will decrease the distance between \( \frac{x}{5} \) and \( \frac{3}{5} \) arbitrarily. Instead, it provides a pattern of interconnected precision between \( \varepsilon \) and \( \delta \). This dependency is what makes limits consistent and defined rigorously across calculus.

Ultimately, the role of epsilon-delta is to ensure precision in the behavior of functions as variables approach specific values, establishing the backbone of continuity and the functioning of calculus.

Limit of a Function

In calculus, the limit of a function as \( x \) approaches a certain value enables us to understand how functions behave near specific points. Using the epsilon-delta definition, this behavior is quantified to confirm and assure proximity in function outputs.

The limit statement \( \lim_{x \rightarrow 3} \frac{x}{5} = \frac{3}{5} \) indicates that as \( x \) nears 3, \( \frac{x}{5} \) becomes arbitrarily close to \( \frac{3}{5} \). Considering this through the epsilon-delta lens, we devise a \( \delta \) that ensures this closeness, demonstrating that even at infinitesimally small intervals around 3, the function matter-of-factly approaches \( \frac{3}{5} \).

Applications of limits of a function include:

• Establishing continuity and differentiability, basics for derivative calculations.
• Supporting integral calculi, crucial for area-functions analysis.
• Enhancing approximation methods, like Taylor and Maclaurin series.

Understanding limits through examples like the given exercise allow students to internalize these abstract concepts, hence solidify their foundation in calculus.