Problem 38 - 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. \(\lim _{x \rightarrow a} b=b,\) for any constants \(a\) and \(b\)

Step-by-Step Solution

Answer

Question: Prove that the limit of a constant function is equal to the constant using the epsilon-delta definition of a limit. Answer: To prove that the limit of a constant function is equal to the constant, we considered the epsilon-delta definition. We observed that the left side of the inequality is always 0, meaning no constraint is needed for the relationship between ε and δ. Thus, for any constants a and b, the limit as x approaches a of the constant function f(x) = b is indeed b, written as: lim (x→a) b = b.

Step 1: Write down the epsilon-delta definition of a limit

The epsilon-delta definition of a limit is as follows: $$\forall\varepsilon>0,\ \exists\delta>0,\ \forall x: 0<|x-a|<\delta \Rightarrow |f(x)-L| <\varepsilon$$ In this case, the function \(f(x) = b\) and \(L = b\). Our goal is to find an appropriate relationship between \(\varepsilon\) and \(\delta\), which guarantees that \(|b-b| < \varepsilon\) for any \(x\) such that \(0<|x-a|<\delta\).

Step 2: Simplify the left side and establish inequality

Let's simplify the left side of the inequality: $$|b-b| = |0| = 0$$ Since 0 is always less than \(\varepsilon\) for any positive \(\varepsilon\), it means this inequality is always true.

Step 3: Establish relationship between \(\varepsilon\) and \(\delta\)

As we have seen that the inequality is always true, we can say that it is true for any \(\delta\) as well, since there is no constraint on what value \(\delta\) should take. Therefore, the inequality \(0<|x-a|<\delta\) can be satisfied for any relationship between \(\varepsilon\) and \(\delta\). We can simply choose \(\delta>0\) without needing any specific relationship with \(\varepsilon\).

Step 4: Write the conclusion

We have shown that for the function \(f(x) = b\), the limit as \(x\) approaches \(a\) is indeed \(b\): \(\lim _{x \rightarrow a} b = b\). Furthermore, we have shown that there is no constraint on the relationship between \(\varepsilon\) and \(\delta\) since the left side of the inequality is always 0, and therefore the limit holds for any constants \(a\) and \(b\).

Key Concepts

Understanding Limit Proofs

When tackling limit proofs, especially using the epsilon-delta definition, our goal is to show that as we get closer to a specific point, the function approaches a particular value. For the given problem, we are proving that \(\lim_{x \rightarrow a} b = b\), where both \(a\) and \(b\) are constants.

To approach this, we use:

• \(\varepsilon\) (epsilon) as a small positive number representing how close we want to be to the limit.
• \(\delta\) as another small positive number ensuring our proximity to the desired x-value.

The proof aims to create a direct link between these two values to demonstrate the constant behavior of the function.

Handling Constants in Limits

When dealing with limits involving constants, such as \(f(x) = b\), the situation becomes quite straightforward. Here, the function always returns the constant \(b\) regardless of \(x\).

This means:

• No matter how \(x\) changes, as long as \(xeq a\), \(f(x)\) remains \(b\).
• The difference \(|b-b|\) is always 0, making the inequality trivial.

This simplicity is crucial for understanding why proofs involving constants can appear almost effortless. Here, the function's limit equals the constant itself without needing additional complex calculations.

Connecting Epsilon and Delta

The relationship between \(\varepsilon\) and \(\delta\) forms the heart of limit proofs. It ensures the function stays close to its limit when \(x\) nears \(a\). In our case, because \(|b-b| = 0\), we initially might think there's no direct need for a connection.

However, here:

• Since \(0 < \varepsilon\), it automatically satisfies \(|b-b| < \varepsilon\).
• This lack of dependency means any positive \(\delta\) works regardless of \(\varepsilon\).

Thus, proving the limit becomes simpler, emphasizing that the constants have inherently stable behavior independent of \(\varepsilon\) and \(\delta\).

Mastering Mathematical Inequalities

In limit proofs, mathematical inequalities help formalize our arguments. They ensure the function’s output stays within the desired bounds as input approaches a particular value. For \(f(x) = b\), the key step involved showing:

• \(|b-b| = 0\)
• This inequality holds for any \(\varepsilon > 0\), ensuring the proof always works.

Understanding these inequalities allows us to establish why limits behave as they do, confirming our proofs' validity without further complicated steps.
By mastering these inequalities, you'll appreciate the seamless nature of working with constants and see how they simplify our understanding of limits. This knowledge offers a powerful tool in any calculus student's toolkit.