Problem 14 - Calculus: Early Transcendentals

Question

Given that \( \displaystyle \lim_{x \to 2}(5x - 7) = 3 \), illustrate Definition 2 by finding values of \( \delta \) that correspond to \( \varepsilon = 0.1 \), \( \varepsilon = 0.05 \), and \( \varepsilon = 0.01 \).

Step-by-Step Solution

Answer

For \( \varepsilon = 0.1 \), \( \delta = 0.02 \); for \( \varepsilon = 0.05 \), \( \delta = 0.01 \); for \( \varepsilon = 0.01 \), \( \delta = 0.002 \).

Step 1: Understand the Definition of Limit

The definition states that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 2| < \delta \), then \(|(5x - 7) - 3| < \varepsilon \). We need to find corresponding \( \delta \) values for given \( \varepsilon \) values.

Step 2: Rewrite the Limit Condition

Start with \( |(5x - 7) - 3| < \varepsilon \). Simplify it to \( |5x - 10| < \varepsilon \). This implies \( 5|x - 2| < \varepsilon \).

Step 3: Solve for \( |x - 2| \)

Divide both sides of \( 5|x - 2| < \varepsilon \) by 5 to find \( |x - 2| < \frac{\varepsilon}{5} \). The expression \( \frac{\varepsilon}{5} \) gives the corresponding \( \delta \) value for a specific \( \varepsilon \).

Step 4: Solve for Specific \( \varepsilon \) Values

Now apply the formula from Step 3 to each \( \varepsilon \):- If \( \varepsilon = 0.1 \), then \( \delta = \frac{0.1}{5} = 0.02 \).- If \( \varepsilon = 0.05 \), then \( \delta = \frac{0.05}{5} = 0.01 \).- If \( \varepsilon = 0.01 \), then \( \delta = \frac{0.01}{5} = 0.002 \).

Key Concepts

Limit Proof

Understanding how to prove a limit is essential in calculus. A limit proof shows that as a variable approaches a specific value, a function gets closer to a particular number. It's like proving that as you walk toward a destination, you're getting closer to it step by step. The epsilon-delta proof is a standard method used in this process.

Proving a limit involves several steps:

• Identify the function and the point as the variable approaches.
• Formulate what it means to get 'close enough.' This involves the quantities epsilon (\(\varepsilon\)) and delta (\(\delta\)).
• Show that for every \(\varepsilon > 0\), we can find a \(\delta > 0\), the condition \( 0 < |x - 2| < \delta \) results in \(|f(x) - L| < \varepsilon\), where \(L\) is the limit value.
• Work through the specific example, determining which \(\delta\) values fit the given \(\varepsilon\) conditions.

Starting from this conceptual framework, you can derive \(\delta\) values that prove a function reaches its limit at a point. This kind of proof demonstrates mathematical rigour and accuracy in understanding limits.

Epsilon-Delta Criterion

The epsilon-delta criterion is at the heart of understanding the formal definition of a limit. This method might seem complex initially, but it's a powerful way to confirm that a function approaches a limit. Here's a simpler breakdown:

- **Epsilon (\(\varepsilon\))** represents how close the function's output (or \( f(x) \)) needs to be to the limit value.
- **Delta (\(\delta\))** corresponds to how near the input \(x\) must be to the target value to meet the epsilon condition.

To utilize the epsilon-delta criterion:

• Begin by expressing the desired closeness of the function to the limit, given by \(|f(x) - L| < \varepsilon\).
• Determine a distance \(\delta\) for \(x\) so that \(0 < |x - c| < \delta\) enforces the inequality above.

In our case, the function is \(f(x) = 5x - 7\), with a limit \(L = 3\) as \(x \to 2\). The steps reveal how to manipulate this inequality to find the required \(\delta\) values for each \(\varepsilon\) value. Finally, choosing \(\delta = \frac{\varepsilon}{5}\) perfectly bridges these math conditions. This epsilon-delta approach is a cornerstone in calculus for rigorously examining limits.

Calculus Limits

Limits form the foundational pillar in calculus, essential for understanding concepts like continuity, derivatives, and integrals. By grasping limits, students unlock a deeper comprehension of calculus as a whole. A limit essentially tells us what happens to a function as inputs get closer and closer to a specified point.

There are several types of limits:

• **Finite Limits**: When a function approaches a specific real number as \(x\) approaches a certain point.
• **Infinite Limits**: Occur when a function grows without bounds as \(x\) approaches a particular value.
• **Limits at Infinity**: Describe the behavior of a function as \(x\) increases or decreases without bound.

In the exercise, you are given a finite limit example where \( \lim_{x \to 2}(5x - 7) = 3 \). We explored how different epsilon values influence the \(\delta\) required. Through comparing these values, it's easier to understand how a function's output gets arbitrarily close to its limit under specific constraints. This kind of exploration enriches the learning process, helping cement the concept of limits as more than just an abstract idea.