Problem 21 - Calculus for Biology and Medicine

Question

Use the formal definition of limits to prove each statement. \(\lim _{x \rightarrow c}(m x)=m c\), where \(m\) is a constant

Step-by-Step Solution

Answer

The limit \( \lim_{x \rightarrow c}(mx) = mc \) is proven using \( \delta = \frac{\epsilon}{|m|} \).

Step 1: Understand the Formal Definition of Limit

The formal definition of a limit states that for any \( \epsilon > 0\), there exists a \( \delta > 0\) such that whenever \( 0 < |x - c| < \delta\), it follows that \( |f(x) - L| < \epsilon\). Here, \( L = m c\) and \( f(x) = mx\).

Step 2: Set Up the Limit Expression

We want to show that \( |mx - mc| < \epsilon\) whenever \( 0 < |x - c| < \delta\). This involves finding a relationship between \( \epsilon\) and \( \delta\).

Step 3: Simplify the Expression

Consider the expression \( |mx - mc| = |m(x-c)| = |m||x-c|\). We want \( |m||x-c| < \epsilon\).

Step 4: Determine the Appropriate \(\delta\)

Since \( |m||x-c| < \epsilon\), we can divide both sides by \( |m| \) (assuming \( m eq 0 \)) to get \( |x-c| < \frac{\epsilon}{|m|} \). Thus, choose \( \delta = \frac{\epsilon}{|m|} \).

Step 5: Conclude the Proof

With \( \delta = \frac{\epsilon}{|m|} \), for any \( \epsilon > 0 \), if \( 0 < |x - c| < \delta\), then \( |mx - mc| < \epsilon \) holds true. Thus, by the formal definition of limits, \( \lim _{x \rightarrow c}(mx)=mc\).

Key Concepts

Formal Definition of Limits

The formal definition of limits is a foundational concept in calculus, helping to pinpoint the exact behavior of functions as they approach particular values. According to this definition, we say the limit of a function \(f(x)\) as \(x\) approaches \(c\) is \(L\) (written as \( \lim_{x \to c} f(x) = L\)) if for every \( \epsilon > 0\), there exists a \( \delta > 0\) such that whenever \( 0 < |x - c| < \delta\), it follows that \( |f(x) - L| < \epsilon\). In simpler terms, the function \(f(x)\) gets arbitrarily close to \(L\) as \(x\) gets sufficiently near \(c\). This definition is crucial when dealing with continuous functions and requires distinguishing between points arbitrarily close to \(c\), rather than exactly at \(c\). This approach allows calculus to handle functions that might not be well-defined exactly at \(c\). Understanding this framework lays the groundwork for more advanced topics in calculus.

Limit Proof Techniques

When proving limits using the formal definition, a structured approach is key. Let's break down the technique:

• **Expression Simplification:** Begin by setting up the expression \( |f(x) - L| < \epsilon\). Simplify this expression to form a more manageable equation.
• **Epsilon-Delta Relationship:** The main task is to express \( \epsilon\) and \( \delta\) in relation to each other. By manipulating the inequality obtained from the simplified expression, find a suitable \( \delta\) for a given \( \epsilon\).
• **Choosing Delta:** Determine \( \delta\) that serves as a sufficient constraint on \(x\) to ensure the inequality holds true. Often this involves dividing \( \epsilon\) by known constants from the simplified expression.
• **Completing the Proof:** Substitute \( \delta\) back into the inequality to check that it holds. This confirms the limit using the formal definition.

These steps are pivotal in analyzing and proving statements about limits, especially for ensuring rigorous mathematical argumentation.

Epsilon-Delta Definition

The epsilon-delta definition is a meticulous mathematical method used to prove statements about limits. It is essential for rigorously establishing how a function behaves near a particular point. Here is how it works:

• Epsilon (\(\epsilon\)): Represents how close \( f(x)\) needs to be to \(L\), the limiting value. It is typically a small positive number that showcases the precision with which \(f(x)\) approximates \(L\).
• Delta (\(\delta\)): Represents how close \(x\) must be to \(c\) to keep \(f(x)\) within \(\epsilon\) of \(L\). For every \(\epsilon\), \(\delta\) is determined to establish this proximity.

This definition enables precise articulation of the intuitive idea of 'approaching'. It is central to confirming that a limit exists and is equal to some specified value. The combination of \(\epsilon\) and \(\delta\) through inequality ensures that all scenarios where \( x\) nears \( c\) keep \( f(x)\) within range of \( L\). This structured method anchors calculus' ability to deal with the concept of continuity and change at a precise level.