Question

In Exerises 2.1.28-2.1.30 consider only the case where at least one of \(L_{1}\) and \(L_{2}\) is \(\pm \infty\). Prove: If \(\lim _{x \rightarrow x_{0}} f(x)=L_{1}, \lim _{x \rightarrow x_{0}} g(x)=L_{2},\) and \(L_{1}+L_{2}\) is not indetermi- nate, then $$ \lim _{x \rightarrow x_{0}}(f+g)(x)=L_{1}+L_{2} $$

Short answer

Question: Prove that if at least one of the limits \(L_1\) and \(L_2\) is \(\pm \infty\) and the sum of the limits is not indeterminate, then \(\lim_{x\to x_0}(f+g)(x) = L_1 + L_2\). Answer: This statement has been proven using the limit sum rule and by analyzing the possible cases for \(L_1\) and \(L_2\). When at least one of the limits is \(\pm \infty\) and their sum is not indeterminate, we have established that the limit of the sum of the functions is equal to the sum of the limits, i.e., \(\lim_{x\to x_0}(f+g)(x) = L_1 + L_2\).

Step-by-step solution

Use the limit sum rule

The limit sum rule states that if \(\lim_{x\to a} f(x)=L_1\) and \(\lim_{x\to a} g(x)=L_2\), then \(\lim_{x\to a} (f+g)(x) = L_1+L_2.\) We need to show that this rule holds for the given conditions.

Determine possible cases for \(L_1\) and \(L_2\)

Since at least one of \(L_1\) and \(L_2\) is \(\pm \infty\), we have three possible cases: 1. \(L_1=\infty\) and \(L_2=-\infty\) 2. \(L_1=\infty\) and \(L_2\in\mathbb{R}\) 3. \(L_1\in\mathbb{R}\) and \(L_2=\infty\) We are given that \(L_1+L_2\) is not indeterminate, so we can exclude the case of \(L_1=\infty\) and \(L_2=-\infty\) since this case would yield an indeterminate form.

Prove the limit sum for other cases

We now need to prove the sum of the limits for the cases where \(L_1+L_2\) is not indeterminate: Case 1: \(L_1=\infty\) and \(L_2\in\mathbb{R}\) Using the limit sum rule, we have: \(\lim_{x\to x_0}(f+g)(x) = \lim_{x\to x_0} f(x) + \lim_{x\to x_0} g(x) =\infty + L_2 = \infty = L_1 + L_2\) Case 2: \(L_1\in\mathbb{R}\) and \(L_2=\infty\) Using the limit sum rule, we have: \(\lim_{x\to x_0}(f+g)(x) = \lim_{x\to x_0} f(x) + \lim_{x\to x_0} g(x) = L_1 + \infty = \infty = L_1 + L_2\) Since the sum of the limits is equal to \(L_1+L_2\) for the cases considered, we have proved that if at least one of \(L_1\) and \(L_2\) is \(\pm \infty\) and \(L_1+L_2\) is not indeterminate, then \(\lim_{x\to x_0}(f+g)(x) = L_1 + L_2\).

Key concepts

Infinite Limits

Infinite limits occur when the value of a function grows without bound as the input approaches a certain point. This might mean that the function increases indefinitely toward positive infinity, or it can decrease without limit toward negative infinity.

Understanding infinite limits is crucial because it helps in determining the behavior of functions at points where they are not well-defined. It tells us how the function behaves near these points, despite not reaching a finite value.

When dealing with infinite limits, remember that:

• If the limit is positive infinity, the function's values become larger and larger positive numbers.
• If the limit is negative infinity, the function's values become larger and larger negative numbers.

Infinite limits are often encountered in rational functions or when dealing with vertical asymptotes on a graph. Recognizing infinite limits helps in understanding and simplifying complex expressions, especially as we approach certain critical points where the function tends to spike or dip sharply.

Limit Sum Rule

The limit sum rule is a fundamental concept in calculus that simplifies evaluating limits of sums of functions. Simply put, if you have two functions, say \(f(x)\) and \(g(x)\), and you know their limits as \(x\) approaches a certain value, you can find the limit of their sum easily.

This rule states that:

• If \(\lim_{x\to a} f(x) = L_1\) and \(\lim_{x\to a} g(x) = L_2\), then \(\lim_{x\to a} (f(x)+g(x)) = L_1 + L_2\).

This rule assumes that the limits \(L_1\) and \(L_2\) exist and are either finite or infinite in a non-indeterminate way.

Applying the limit sum rule is straightforward, making complex limit evaluations manageable by breaking them into simple parts. By combining the known limits, you can easily determine the result of the limit of a sum of functions.

It's an especially useful tool when functions exhibit infinite behavior, as it can confirm whether sums of infinite limits still yield coherent mathematical results or not.

Indeterminate Forms

Indeterminate forms emerge in calculus when mathematical expressions do not directly lead to a clear limit. These forms are typically seen as "0/0" or "∞-∞," where standard operations do not apply effectively.

Indeterminate forms require careful analysis, often using methods such as L'Hôpital's Rule or algebraic manipulation to resolve the uncertainty and find the actual limit.

Key indeterminate forms include:

• \(\frac{0}{0}\)
• \(\infty - \infty\)
• \(0 \cdot \infty\)
• \(\text{and more}\)

Each form needs a specific approach or strategy to handle and interpret it correctly.

By recognizing and addressing indeterminate forms correctly, we can ensure that the mathematical operations lead to meaningful and accurate results. It’s crucial when the expression under analysis is complex or contains terms that flare out to infinity. Understanding and resolving these forms are integral for efficiency in calculus operations and for avoiding common mistakes in limit calculations.