Problem 36 - Calculus Early Transcendentals

Question

Prove that $$\lim _{x \rightarrow 2} \frac{1}{x}=\frac{1}{2}$$

Step-by-Step Solution

Answer

Yes, the limit \( \lim_{x \to 2} \frac{1}{x} = \frac{1}{2} \) is proven by the epsilon-delta definition.

Step 1: Understand the limit definition

The expression \( \lim_{x \to a} f(x) = L \) means that as \( x \) approaches \( a \), the function \( f(x) \) approaches the limit \( L \). In this exercise, as \( x \) approaches \( 2 \), \( \frac{1}{x} \) approaches \( \frac{1}{2} \).

Step 2: Define the epsilon-delta criterion

According to the epsilon-delta definition of a limit, we say \( \lim_{x \to 2} \frac{1}{x} = \frac{1}{2} \) if for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 2| < \delta \), then \( \left| \frac{1}{x} - \frac{1}{2} \right| < \varepsilon \).

Step 3: Find an expression for \( \left| \frac{1}{x} - \frac{1}{2} \right| \)

Start by simplifying the expression \( \left| \frac{1}{x} - \frac{1}{2} \right| \). You can rewrite it as:\[ \left| \frac{1}{x} - \frac{1}{2} \right| = \left| \frac{2 - x}{2x} \right|. \]

Step 4: Ensure bounding of the expression

We need to bound \( \left| \frac{2-x}{2x} \right| \) such that it is less than \( \varepsilon \). Assume \( |x - 2| < \delta \). This means \( 1.9 < x < 2.1 \), ensuring \( x \) is close to 2 but not equal to zero.

Step 5: Choose \( \delta \) based on \( \varepsilon \)

To ensure \( \left| \frac{2-x}{2x} \right| < \varepsilon \), select \( \delta \leq 0.1 \) so that \( 2x \approx 4 \) when \( x \approx 2 \). Therefore, choose \( \delta = \min \left( 0.1, \frac{2\varepsilon}{5} \right) \).

Step 6: Validate the epsilon-delta inequality

Substitute \( \delta \) into the expressions, giving: \[ \left| \frac{2-x}{2x} \right| < \left| \frac{2-x}{4} \right| < \varepsilon.\] This inequality ensures the limit is satisfied, proving that for every \( \varepsilon > 0 \), a corresponding \( \delta \) can be selected.

Key Concepts

Limit Proof

The idea of a limit proof is to demonstrate, through logical reasoning, that a function approaches a certain value as the input nears a specified point. These proofs help us rigorously verify results that might seem obvious intuitively. In this problem, the limit we need to prove is:

• \( \lim_{x \to 2} \frac{1}{x} = \frac{1}{2} \)

To embark on a limit proof, you'll start by understanding what it means for a function to have a limit at a particular point. When working through a proof, typically you'll:

• Clarify the goal using the limit definition.
• Apply the epsilon-delta criteria to establish bounds.
• Pick appropriate values for delta based on epsilon to satisfy the inequality.

Limit proofs require a systematic approach and often involve algebraic manipulation to bound expressions in terms of epsilon and delta.

Epsilon-Delta Criterion

The epsilon-delta criterion is a formal way to define the limit of a function. Here, the primary concept revolves around two very small positive numbers called epsilon (\( \varepsilon \)) and delta (\( \delta \)).

• Epsilon (\( \varepsilon \)) represents the maximum allowable difference between the function's value and the limit.
• Delta (\( \delta \)) relates to how tightly the input values must cluster around the point of approach to maintain that difference lower than epsilon.

This definition guarantees that no matter how small an error margin (epsilon) you choose, it's possible to find a delta which corresponds to it, such that if the input is within delta of the point, the function's output stays within epsilon of the limit value.
Our specific task is to ensure:

• If \( 0 < |x - 2| < \delta \), then \( \left| \frac{1}{x} - \frac{1}{2} \right| < \varepsilon \)

This criterion assures us that as \( x \) draws closer to 2, \( \frac{1}{x} \) will indeed get nearer to \( \frac{1}{2} \).

Limit of a Function

A limit describes the behavior of a function as its input approaches a certain value. It doesn't necessarily have to be the function's value at that point, but rather the value the function tends towards.
Limits are foundational in calculus, embodying the idea of approaching a point continuously. Understanding limits sequentially helps grasp the concept better.

• First, think about the function's behavior, like \( \frac{1}{x} \) when \( x \) is near but not equal to 2.
• Consider what happens to \( \frac{1}{x} \) as \( x \) grows exceedingly close to 2. The closer \( x \) is to 2, \( \frac{1}{x} \) approaches 0.5.

The goal is to solidify this tendency as an estimated mathematical fact using the epsilon-delta definition, proving rigorously why \( \frac{1}{2} \) is the limiting value. This is an essential part of ensuring the behavior of functions is predictable and consistent under rigorous scrutiny.