Problem 28 - Calculus: Early Transcendentals

Question

Prove the statement using the \( \varepsilon \), \( \delta \) definition of a limit. \( \displaystyle \lim_{x \to -6^+} \sqrt[5]{6 + x} = 0 \)

Step-by-Step Solution

Answer

The limit is proven by showing \\( 0 < \sqrt[5]{6 + x} < \varepsilon \\) for \\( \delta = \varepsilon^5 \\).

Step 1: Identify Limit Definition

The \( \varepsilon \, \delta \) definition of a limit states that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < x + 6 < \delta \, \) then \(|\sqrt[5]{6 + x} - 0| < \varepsilon\). Our goal is to show this holds true for \( x \) approaching \(-6^+\).

Step 2: Relate Function to Epsilon

Since \( \lim_{x \to -6^+} \sqrt[5]{6 + x} = 0 \), we want \( |\sqrt[5]{6 + x}| < \varepsilon \). This simplifies to \( \sqrt[5]{6 + x} < \varepsilon \) since \( \sqrt[5]{6 + x} \) is positive for \( x \to -6^+ \).

Step 3: Express Delta in Terms of Epsilon

To satisfy \( |\sqrt[5]{6 + x}| < \varepsilon \, \) we take \( \sqrt[5]{6 + x} < \varepsilon \. \) This implies \( 6 + x < \varepsilon^5 \. \) Solving for \( x \) gives \( x < \varepsilon^5 - 6 \. \) Choose \( \delta = \varepsilon^5\).

Step 4: Validate the Choice of Delta

Assume \( 0 < x + 6 < \delta = \varepsilon^5 \. \) Then \( 0 < \sqrt[5]{6 + x} < \varepsilon \. \) This ensures that \( |\sqrt[5]{6 + x} - 0| < \varepsilon \, \) thus fulfilling the definition of the limit.

Key Concepts

Limit of a Function

When dealing with the limit of a function, we are interested in understanding how a function behaves as the input approaches a particular value. In this case, the focus is on the limit of the function \( f(x) = \sqrt[5]{6 + x} \) as \( x \) approaches \( -6^+ \):

• A limit can be intuitively thought of as the value a function "approaches" as the input gets arbitrarily close to a given point.
• For the specific function \( \sqrt[5]{6 + x} \), as \( x \) approaches \(-6^+\), the function approaches \( 0 \).

To rigorously show this idea using the epsilon-delta definition, we aim to prove under any given \( \varepsilon > 0 \) (arbitrary small number), there exists a \( \delta > 0 \) such that if \( x \) is within \( \delta \) of \( -6 \), then \( f(x) \) is within \( \varepsilon \) of 0. This formalism provides a precise method for verifying the behavior of limits.

Real Analysis

Real analysis provides the rigorous foundation for studying limits, continuity, and other fundamental concepts in calculus. It employs precise definitions and logical reasoning to delve into the behavior of real-valued functions. The epsilon-delta definition is a cornerstone concept here:

• The \( \varepsilon \), \( \delta \) method is essential for establishing the existence of a limit, offering a formal way to express the approach of a function towards a specific value.
• It relies on creating a "neighborhood" around the point of interest, ensuring the function value stays within a prescribed vicinity, called \( \varepsilon \), by controlling \( \delta \), the distance in the domain.

In proving limits, particularly in real analysis, such rigorous methods transform intuitive concepts into ones we can prove logically, strengthening our understanding of calculus as a robust mathematical discipline.

Calculus Proofs

Calculus proofs often require the use of structured reasoning and logical arguments to establish mathematical truths. With the epsilon-delta definition of limits, the proof process involves:

• Clearly stating the conditions of the problem, such as the input value \( x \) approaching \(-6^+\) in our case.
• Determining transformations of the conditions into manageable mathematical expressions, like expressing \( \delta \) in terms of \( \varepsilon \).
• Validating that the transformations hold under the defined constraints, confirming that the function's output meets the given criteria.

Through calculus proofs, such as in our case study, we not only solve for specific limits but also practice the application of rigorous mathematical techniques essential for advanced problem-solving. These steps are foundational for building a deep understanding of continuous functions and their properties.