Problem 14 - Single Variable Calculus

Question

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow-2}(2 x-3)=-7\)

Step-by-Step Solution

Answer

Given \(\epsilon > 0\), let \(\delta = \frac{\epsilon}{2}\). If \(0 < |x + 2| < \delta\), then \(2|x + 2| < \epsilon\), which satisfies the limit definition since \(|2x + 4| = 2|x + 2|\). Therefore, we have proved that \(\lim_{x \rightarrow -2}(2x - 3) = -7\).

Step 2: Simplify the expressions

First, simplify the expressions inside the absolute value signs: 1. \(|x - (-2)| = |x + 2|\) 2. \((2x - 3) - (-7) = 2x - 3 + 7 = 2x + 4\) Now, the condition is: If \(0 < |x + 2| < \delta\), then \(|2x + 4| < \epsilon\).

Step 3: Relate the expressions

In order to find an appropriate \(\delta\) in terms of \(\epsilon\), we need to relate the two absolute value expressions. To do this, we will try to represent \(|2x + 4|\) in terms of \(|x + 2|\). Factor 2 from \(|2x + 4|\) to get \(2|x + 2|\).

Step 4: Find the value of delta

Now, we want to find the value of \(\delta\) such that: If \(0 < |x + 2| < \delta\), then \(2|x + 2| < \epsilon\). Divide both sides of the inequality by 2: \(|x + 2| < \frac{\epsilon}{2}\). Now notice that if we choose \(\delta = \frac{\epsilon}{2}\), the two inequalities will have the same form, and the limit definition will be satisfied.

Step 5: Verify the result

Let's verify that the value of \(\delta\) we found satisfies the limit definition. Given \(\epsilon > 0\), let \(\delta = \frac{\epsilon}{2}\). If \(0 < |x + 2| < \delta\), then: \(2|x + 2| < 2\delta\) \(2|x + 2| < 2\left(\frac{\epsilon}{2}\right)\) \(2|x + 2| < \epsilon\). Which is the desired inequality, since \(|2x + 4| = 2|x + 2|\). Therefore, based on the precise definition of a limit, we have proved that: \[ \lim_{x \rightarrow -2}(2x - 3) = -7 \]

Key Concepts

Precise Definition of a Limit

When we talk about limits in calculus, the precise definition is crucial to understand how functions behave as they approach certain points. The precise definition of a limit provides us with a way to describe intuitively how a function can get arbitrarily close to a specific value as the input approaches a given point. Imagining this with the function \[ \lim_{x \rightarrow -2}(2x - 3) = -7 \]means that as \(x\) approaches \(-2\), the function \(2x - 3\) should get very close to \(-7\). In simple terms:

• If you can make \(x\) close enough to \(-2\), \(2x - 3\) will be closer to \(-7\).
• The task is to show this using a precise framework, ensuring mathematical rigour.

It’s a way to solidify the concept of a 'limit' using explicit terms of closeness between function values and their limiting behavior.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to prove statements about limits, and it uses two small values: \(\epsilon\) and \(\delta\). Here's the basic idea:

• \(\epsilon\) represents how close we want \(2x - 3\) to be to \(-7\) (or any limit value).
• \(\delta\) represents a small margin around \(x = -2\).

The epsilon-delta method asks us to:

• Find a \(\delta\) such that whenever the distance between \(x\) and \(-2\) is less than \(\delta\),
• Then, the distance between \(2x - 3\) and \(-7\) will be less than \(\epsilon\).

To apply it to our function, we calculated that if \(0 < |x + 2| < \delta\), then \(|2x + 4| < \epsilon\). Choosing \(\delta = \frac{\epsilon}{2}\) works because multiplying it back by 2 gives the right bound upon which \(|2x + 4| < \epsilon\) consistently holds. This process ensures the function value is as close to the limit as specified by the \(\epsilon\).

Limit Proof Methods

There are several methods to prove that the limit of a given function reaches a specific value as the input approaches a particular point. Understanding these methods can help students to choose the most effective approach in different situations.The epsilon-delta method is among the most fundamental because:

• It visualizes all aspects of a limit using specific numbers, meaning the process can be applied to any continuous function.
• It verifies that every intended mathematical statement about a limit can be stated rigorously and proven without ambiguity.

However, analyzing algebraic manipulation helps in simplifying the steps to prove a limit. As seen in the exercise, expressing terms appropriately by factoring out constants simplifies postulates, making relationship derivations between \(\epsilon\) and \(\delta\) more intuitive.These steps confirm that the sought limit is practically achievable and verifiable. The proving process involves simplifying the function to reveal these intrinsic relationships directly. Applying such methods can provide confidence that limits are well-understood and well-argued.