Problem 31 - Calculus Early Transcendentals

Question

Prove the statement using the \(\varepsilon\) , \(\delta\) definition of limit. \(\lim _{x \rightarrow-2}\left(x^{2}-1\right)=3\)

Step-by-Step Solution

Answer

The limit is proved by making \(|x^2 - 4| < \varepsilon\) for \(\delta = \min(1, \frac{\varepsilon}{5})\).

Step 1: Understanding the \\(\varepsilon, \delta\\) Definition of Limit

To prove \(\lim_{x \rightarrow -2} (x^2 - 1) = 3\) using the \(\varepsilon, \delta\) definition, we need to show that for every \(\varepsilon > 0\), there exists some \(\delta > 0\) such that if \(0 < |x + 2| < \delta\), then \(|(x^2 - 1) - 3| < \varepsilon\).

Step 2: Simplifying the Expression

We start by simplifying \(|(x^2 - 1) - 3|\): \[|(x^2 - 1) - 3| = |x^2 - 4|\]This is the expression we want to make less than \(\varepsilon\).

Step 3: Factoring the Expression

Notice that \(|x^2 - 4| = |(x + 2)(x - 2)|\). Our goal is to establish a relation between this expression and \(\varepsilon\).

Step 4: Bounding \(|x - 2|\)

Assume \(|x + 2| < 1\). This implies that \(-3 < x < -1\). Thus, \(-3 < x < -1\) implies \(-5 < x - 2 < -3\) so \(|x - 2| < 5\).

Step 5: Choosing \(\delta\)

Now we need \(|(x + 2)(x - 2)| < \varepsilon\). Since \(|x - 2| < 5\), we have:\[|(x + 2)(x - 2)| < |x + 2| \times 5 < \varepsilon\]We choose \(\delta = \min\left(1, \frac{\varepsilon}{5}\right)\).

Step 6: Verifying the Choice of \(\delta\)

With this choice of \(\delta\), if \(|x + 2| < \delta\), then \(|x^2 - 4| < \varepsilon\) as:\[|x^2 - 4| = |(x + 2)(x - 2)| < 5 \times \frac{\varepsilon}{5} = \varepsilon\]Thus, we have shown the limit according to the \(\varepsilon, \delta\) definition.

Key Concepts

Delta-Epsilon Proof

Delta-epsilon proof is a rigorous method used in calculus to define the concept of limits. It's essential for understanding how functions behave as they approach a particular point.

This proof revolves around two small numbers: \(\varepsilon\) and \(\delta\).

• \(\varepsilon\) represents how close we want the function's value to be to the limit.
• \(\delta\) is about how close \(x\) should be to the point of interest for the function value to remain within the \(\varepsilon\) distance of the limit.

In using the epsilon-delta method, our goal is to find a \(\delta\) for every \(\varepsilon\) that satisfies the limit condition. If we can show that this condition holds, then we've proven the limit using this method. This process ensures the function approaches its limit in a controlled manner.

Limits in Calculus

Limits in calculus are foundational concepts that help us understand the behavior of functions as they approach a specific point. They are crucial for defining derivatives and integrals, which are the core operations in calculus.

Understanding limits involves imagining the value a function reaches as the input approaches a particular point, even if the function does not exactly reach that value.

• Limits deal with the "approaching" of a value rather than reaching it precisely.
• They help in dealing with "undefined" situations, like division by zero.

The notation \(\lim_{x \to a} f(x)\) indicates that we observe how \(f(x)\) behaves as \(x\) gets closer to \(a\). Calculating limits allows us to predict the behavior of functions in a more nuanced way, making calculations more precise.

Continuity and Limits

Continuity in calculus is related to limits in that it describes how a function behaves without any breaks or jumps within its domain. For a function to be continuous at a point, its limit must exist, and this limit should be equal to the function's value at that point.

In mathematical terms, a function \(f(x)\) is continuous at \(x = a\) if:

• The limit \(\lim_{x \to a} f(x)\) exists.
• \(f(a)\) is defined (the function has a value at \(x = a\)).
• \(\lim_{x \to a} f(x) = f(a)\).

Practically, this means you can draw the graph of a continuous function without lifting your pencil from the paper. The epsilon-delta definition is a tool to rigorously prove this continuity by verifying conditions of limits.

Calculus Theorems

Calculus theorems provide the rules and foundations for the amazing results we achieve in calculus. They are fundamental to the study of mathematics and enable mathematicians to deduce new results and understand existing ones.

Here are a few essential calculus theorems related to limits and continuity:

• The Squeeze Theorem: Useful when you can "squeeze" a function between two others that have the same limit at a point.
• The Intermediate Value Theorem: States that if a function is continuous over an interval, it takes every value between \(f(a)\) and \(f(b)\).
• The Limit Laws: Provide rules for addition, subtraction, multiplication, etc., of limits.

These theorems provide structured approaches to solving calculus problems, ensuring that mathematicians can derive accurate and consistent results efficiently.