Problem 30 - Calculus: Early Transcendentals

Question

Prove the statement using the \( \varepsilon \), \( \delta \) definition of a limit. \( \displaystyle \lim_{x \to 2} (x^2 + 2x - 7) = 1 \)

Step-by-Step Solution

Answer

The limit is proved using the \( \varepsilon, \delta \) definition; \( \delta = \min(1, \frac{\varepsilon}{7}) \).

Step 1: Understand the definition

The definition states: For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \), if \( 0 < |x - 2| < \delta \) then \( |f(x) - 1| < \varepsilon \). We need to find such a \( \delta \) for the function \( f(x) = x^2 + 2x - 7 \).

Step 2: Simplify the expression

Let's simplify \( |f(x) - 1| \). We have \( f(x) = x^2 + 2x - 7 \). Thus, \( f(x) - 1 = x^2 + 2x - 7 - 1 = x^2 + 2x - 8 \). So, we need \( |x^2 + 2x - 8| < \varepsilon \).

Step 3: Factor and find bounds

To factor \( x^2 + 2x - 8 \), we solve: \( (x - 2)(x + 4) \). Therefore, \( |x^2 + 2x - 8| = |(x - 2)(x + 4)| \). We want \( |(x-2)(x+4)| < \varepsilon \). If \( |x - 2| < 1 \), then \( 1 < x < 3 \), so \( 5 < x + 4 < 7 \), and hence \( |x + 4| < 7 \).

Step 4: Determine \( \delta \) in terms of \( \varepsilon \)

We have \( |(x-2)(x+4)| = |x-2||x+4| < \varepsilon \). Given \( |x+4| < 7 \), we have \( |x-2| \cdot 7 < \varepsilon \). Therefore, \( |x-2| < \frac{\varepsilon}{7} \). Choose \( \delta = \min(1, \frac{\varepsilon}{7}) \).

Step 5: Conclusion of the proof

For \( \varepsilon > 0 \), we chose \( \delta = \min(1, \frac{\varepsilon}{7}) \). If \( 0 < |x - 2| < \delta \), then \( |x^2 + 2x - 8| < \varepsilon \). Thus, by the \( \varepsilon, \delta \) definition of a limit, we have \( \lim_{x \to 2} (x^2 + 2x - 7) = 1 \).

Key Concepts

Limit Proofs

Understanding limit proofs, especially using the epsilon-delta definition, is crucial in calculus. These proofs show that as a variable approaches a particular value, a function tends to a specific limit. The epsilon-delta definition is a precise way to define limits and involves two main components:

• **Epsilon (\( \varepsilon \))**: A small positive number representing how close we want the function's value to be to the limit.
• **Delta (\( \delta \))**: A small positive number indicating how close input values (\( x \)) must be to the desired point (without being equal to it) to ensure the function's value stays within the epsilon range of the limit.

When proving limits, we need to:

• Start by examining the condition: For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \).
• Ensure that for all \( x \), when \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \varepsilon \).

In the given example, the challenge was to find the appropriate delta that satisfies this relationship for a polynomial. By simplifying and manipulating the expression, we ensured that the function’s value is as close as desired to the limit as x approaches 2.

Polynomial Functions

Polynomial functions play a significant role in calculus and limit proofs. A polynomial function is one where the right-hand side expression is formed by variables raised to natural number exponents and each term has a constant multiplier. Common examples include:

• Linear functions: e.g. \( ax + b \)
• Quadratic functions: e.g. \( ax^2 + bx + c \)

In the given exercise, we dealt with the quadratic polynomial \( x^2 + 2x - 7 \). A unique characteristic of polynomial functions is their continuity and differentiability over all real numbers. When finding limits with polynomials:

• Identifying factorization opportunities is vital, such as rewriting \( x^2 + 2x - 8 \) as \((x-2)(x+4)\).
• Recognizing patterns, simplifying expressions, and anticipating results can make the solutions more intuitive.

Because of their simple and predictable nature, polynomial functions help affirm limit behaviors through algebraic manipulation, reinforcing the findings with numeric bounds.

Calculus Concepts

In calculus, understanding fundamental concepts like limits, continuity, and differentiability is essential. Limits are at the foundation of calculus and provide insight into a function's behavior as an input value approaches a certain point. For limits:

• They help us understand the function's trend or direction near the point of interest without necessarily reaching the point.
• The epsilon-delta definition concretely measures proximity in terms of predetermined closeness using small positive constants \( \varepsilon \) and \( \delta \).

The exercise illustrates these calculus concepts by showing how limits are not always straightforward or numerically direct. They require a strategic breakdown:

• Simplification and strategic algebraic manipulation gauge how inputs relate to outputs as they near specific points.
• In calculus, this breakdown aids in understanding how continuous functions behave and remain unidirectional within defined bounds.
• This understanding empowers us to handle more complex scenarios and functions through calculated steps and predictable outcomes.

Overall, these calculus concepts provide a framework not just for solving equations but also for deeper explorations of mathematical analysis and behaviors.