Problem 24 - Calculus: Early Transcendentals

Question

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between \(\varepsilon\) and \(\delta\) that guarantees the limit exists. $$\lim _{x \rightarrow 0}|5 x|=0$$

Step-by-Step Solution

Answer

Question: Prove the limit $$\lim _{x \rightarrow 0}|5 x|=0$$ using the precise definition of a limit. Answer: To prove the limit, we found a relationship between 𝜖 and 𝛿 such that if we choose 𝛿 = 𝜖/5, it guarantees that for any given 𝜖 > 0, there exists a 𝛿 > 0 such that if 0 < |x| < 𝛿, then |5x| < 𝜖. This satisfies the precise definition of a limit, and therefore, we have proven that $$\lim _{x \rightarrow 0}|5 x|=0$$ with the relationship 𝛿 = 𝜖/5.

Step 1: State the given limit

We are given the limit $$\lim _{x \rightarrow 0}|5 x|=0$$ and we need to prove it using the precise definition of a limit. Step 2: Write down the precise definition of a limit

Step 2: State the precise definition of a limit

For any given \(\varepsilon > 0\), there must exist a \(\delta > 0\) such that if \(0 < |x - 0| < \delta\), then \(| |5x| - 0 | < \varepsilon\). Step 3: Manipulate the expression

Step 3: Simplify the given expression

Since \(|5x|\) is always non-negative, we can write the expression as \(| |5x| - 0 | = |5x|\). Step 4: Find relationship between \(\varepsilon\) and \(\delta\)

Step 4: Determine the relationship between \(\varepsilon\) and \(\delta\)

If \(0 < |x| < \delta\), then \(|5x| = 5|x| < 5\delta\). We want to find a relationship between \(\varepsilon\) and \(\delta\) such that \(|5x| < \varepsilon\). Since \(\varepsilon > 0\), we can choose \(\delta = \frac{\varepsilon}{5}\). This ensures that if \(0 < |x| < \delta\), then \(|5x| < 5\delta = 5\cdot \frac{\varepsilon}{5} = \varepsilon\). Step 5: Prove the limit

Step 5: Prove the limit using the relationship between \(\varepsilon\) and \(\delta\)

For any \(\varepsilon > 0\), if we choose \(\delta = \frac{\varepsilon}{5}\), we have that \(0 < |x| < \delta \Rightarrow |5x| < \varepsilon\). This satisfies the precise definition of a limit, so we have proven that $$\lim _{x \rightarrow 0}|5 x|=0$$ with the relationship \(\delta = \frac{\varepsilon}{5}\).

Key Concepts

Precise Definition of a Limit

The precise definition of a limit is a fundamental concept in calculus. Essentially, it helps to understand how a function behaves as it approaches a specific point. In formal terms, the limit of a function \(f(x)\) as \(x\) approaches a certain value \(c\) is \(L\), if for every positive number \(\varepsilon\) (no matter how small), there is another positive number \(\delta\) such that whenever \(|x - c| < \delta\), it implies that \(|f(x) - L| < \varepsilon\).

In simpler words, no matter how close you want \(f(x)\) to get to \(L\), you can always make \(x\) close enough to \(c\) to achieve this closeness within a distance \(\varepsilon\). This is what makes the concept of a limit "precise" and helps in rigorous proofs.

This definition forms the backbone of understanding limits and is crucial for proving more complex mathematical theorems.

Epsilon-Delta Definition

The epsilon-delta definition is essentially a technical tool that gives us a framework to prove limits. It might seem complex initially, but understanding the mechanics is key. When you hear "epsilon-delta," think about making things arbitrarily small.

Here's how it works:

• \(\varepsilon\) (epsilon) represents how close you'd like \(f(x)\) to be to the limit \(L\).
• \(\delta\) (delta) represents a range of values for \(x\) that achieve this closeness.

To illustrate, if you want \(|f(x) - L| < \varepsilon\), you find \(\delta\) such that whenever \(|x - c| < \delta\), \(f(x)\) stays within \(\varepsilon\) of \(L\).

In the exercise \( \lim_{x \to 0} |5x| = 0 \), if you pick any \(\varepsilon > 0\), you can choose \(\delta = \frac{\varepsilon}{5}\) ensuring that as \(x\) gets super close to zero, \(|5x|\) becomes smaller than \(\varepsilon\). This methodical approach effectively bridges a conceptual understanding with mathematical rigor.

Limit Concepts

Limit concepts in calculus describe how a function approaches a particular point. These concepts are foundational in understanding whether functions behave predictably in mathematics. When discussing limits, one key point is continuity, which describes a "smooth" path without any breaks or jumps as you follow along the function's graph.

Here are a few important points about limit concepts:

• **Convergence**: This occurs when a function increasingly nears a specific value.
• **Divergence**: Opposite to convergence, this happens when a function veers away from a specific value.
• **Asymptotic Behavior**: Describes how functions behave as they go towards infinity or negative infinity.

In practical terms, limits help calculate instantaneous rates of change, a central theme in derivatives, and they offer insights into the behavior of functions near boundary points. Through the epsilon-delta proof, as demonstrated in the exercise \( \lim_{x \to 0} |5x| = 0 \), we confirm the predictable nature of the function \(|5x|\) approaching zero as \(x\) nears zero.

Understanding these ideas not only clarifies mathematical solutions but also enhances your analytical skills in tackling real-world problems.