Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give a rigorous proof that if \(\lim _{x \rightarrow \infty} f(x)=A\) and \(\lim _{x \rightarrow \infty} g(x)=B\), then $$ \lim _{x \rightarrow \infty}[f(x)+g(x)]=A+B $$

Step-by-Step Solution

Answer

The limit of the sum of functions is \( A + B \) if their individual limits are \( A \) and \( B \).

Step 1: Understanding the Definitions

We are given two functions, \( f(x) \) and \( g(x) \), with limits \( \lim_{x \rightarrow \infty} f(x) = A \) and \( \lim_{x \rightarrow \infty} g(x) = B \). Our goal is to show that the limit of their sum, \( \lim_{x \rightarrow \infty} [f(x) + g(x)] \), is equal to \( A + B \).

Step 2: Setting Up the ε-δ Definition of Limit

By definition, for every \( \epsilon > 0 \), there exists a \( N_1 \) such that for all \( x > N_1 \), \( |f(x) - A| < \frac{\epsilon}{2} \). Similarly, there exists a \( N_2 \) such that for all \( x > N_2 \), \( |g(x) - B| < \frac{\epsilon}{2} \).

Step 3: Choosing an Appropriate δ

Let \( N = \max(N_1, N_2) \). Then, for any \( x > N \), both conditions \( |f(x) - A| < \frac{\epsilon}{2} \) and \( |g(x) - B| < \frac{\epsilon}{2} \) are simultaneously satisfied.

Step 4: Using the Triangle Inequality

Now, consider the expression \( |[f(x) + g(x)] - (A + B)| \). By the triangle inequality, this expression is \( |(f(x) - A) + (g(x) - B)| \leq |f(x) - A| + |g(x) - B| \).

Step 5: Combining Inequalities

Since both \( |f(x) - A| < \frac{\epsilon}{2} \) and \( |g(x) - B| < \frac{\epsilon}{2} \), we have \( |(f(x) - A) + (g(x) - B)| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \).

Step 6: Concluding the Proof

For every \( \epsilon > 0 \), there exists an \( N = \max(N_1, N_2) \) such that for all \( x > N \), \( |[f(x) + g(x)] - (A + B)| < \epsilon \). Thus, by definition of the limit, \( \lim_{x \rightarrow \infty} [f(x) + g(x)] = A + B \).

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a precise way to grasp the concept of limits. It's like setting up a game where you want to get as close as possible to a target. Here, the target is the limit of a function as it approaches a particular value. Imagine you have a number, let's call it \(A\), and you want your function \(f(x)\) to be as close to \(A\) as possible when \(x\) goes to infinity.
For this to happen, the definition states that for every small number \(\epsilon > 0\), which represents the maximum allowed distance from \(A\), you can find a corresponding \(N\) such that when \(x\) is greater than \(N\), the function \(f(x)\) stays within \(\epsilon\) distance of \(A\).
This definition helps provide a rigorous framework for proving limits, ensuring that no matter how small \(\epsilon\) gets, you can always find an \(N\) to satisfy this close relationship. When dealing with sums of functions, like in our original exercise, we apply this definition to each function both \(f(x)\) and \(g(x)\) and combine the results.

Triangle Inequality

The triangle inequality is a fundamental concept in mathematics used to estimate the magnitude of a sum of two numbers or expressions. It suggests that the absolute value of a sum is not greater than the sum of absolute values of the individual parts.
Mathematically, if you have two expressions or numbers \(a\) and \(b\), then the triangle inequality can be expressed as \(|a + b| \leq |a| + |b|\).
In the context of limits, particularly in our original problem, the triangle inequality helps in managing the potential error in our approximations. When dealing with two functions approaching their limits, \(f(x)\) to \(A\) and \(g(x)\) to \(B\), one can state \(|(f(x) - A) + (g(x) - B)| \leq |f(x) - A| + |g(x) - B|\).
This step is crucial because if each part of the inequality is less than small deviations (say \(\epsilon/2\)), it leads us to conclude that their sum can be bounded by \(\epsilon\), completing our proof of the limit of the sum.

Asymptotic Behavior

Asymptotic behavior of a function describes how the function behaves as the variable approaches a certain value, often infinity. It's like understanding the long-term trend of a function, analogous to predicting how a car behaves on an endless road.
In mathematics, particularly in calculus and analysis, recognizing the asymptotic behavior of functions is useful in limit proofs. You're not just interested in what happens at a single point, but what unfolds as you move towards infinity.
For instance, when confirming that \(\lim_{x \to \infty} f(x) = A\), we are describing how \(f(x)\) behaves as \(x\) becomes very large, eventually getting closer and closer to \(A\).
Understanding asymptotic behavior plays a central role in ensuring that our approximations within the epsilon-delta framework remain valid, and it’s a concept that connects closely not only to limits but to various other fields such as algorithms and complex analysis.