Problem 91 - Calculus

Question

Suppose that \(\lim _{x \rightarrow 3} f(x)=7 .\) Are the statements in Problems \(89-95\) true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample. If \(\lim _{x \rightarrow 3} g(x)=5,\) then \(\lim _{x \rightarrow 3}(f(x)+g(x))=12\)

Step-by-Step Solution

Answer

True; using the limit addition rule, the sum is 12.

Step 1: Understand the Given Information

We are given two limit statements: \(\lim_{x \rightarrow 3} f(x) = 7\) and \(\lim_{x \rightarrow 3} g(x) = 5\). These tell us the behavior of the functions \(f(x)\) and \(g(x)\) as \(x\) approaches 3.

Step 2: Apply the Limit Addition Rule

Using the limit addition rule, which states \(\lim_{x \rightarrow c} (f(x) + g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x)\), we find \(\lim_{x \rightarrow 3} (f(x) + g(x)) = 7 + 5\).

Step 3: Compute the Limit

According to the limit addition rule, the limit of the sum is \(7 + 5 = 12\). This tells us that \(\lim_{x \rightarrow 3} (f(x) + g(x)) = 12\).

Step 4: Evaluate the Statement

Since the computed limit \(\lim_{x \rightarrow 3} (f(x) + g(x)) = 12\) matches the expression given in the statement, the statement is true.

Key Concepts

Limit Addition Rule

The Limit Addition Rule is a fundamental concept in calculus used to find the limit of a sum of two functions. This rule states that if the limits of two functions, say \( f(x) \) and \( g(x) \), exist as \( x \) approaches a point \( c \), then the limit of the sum of these functions as \( x \) approaches the same point can be found by simply adding their individual limits. It’s expressed mathematically as:

• \( \lim_{x \rightarrow c} (f(x) + g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x) \)

In the problem, we have \( \lim_{x \rightarrow 3} f(x) = 7 \) and \( \lim_{x \rightarrow 3} g(x) = 5 \). Using the Limit Addition Rule, we add these limits to get:

• \( \lim_{x \rightarrow 3} (f(x) + g(x)) = 7 + 5 = 12 \)

This straightforward addition of the two limits allows us to conclude that the result for the limit of the combined functions is 12.

Behavior of Functions

Understanding the behavior of functions as they approach a specific point is vital in calculus. This behavior is described using limits, which allow us to predict what value a function reaches as the input gets closer to a certain value, without necessarily having the function defined at that point.
In the given exercise, we need to evaluate how the functions \( f(x) \) and \( g(x) \) behave as \( x \) approaches 3. We are provided that \( \lim_{x \rightarrow 3} f(x) = 7 \) and \( \lim_{x \rightarrow 3} g(x) = 5 \). These limits indicate that, regardless of whether \( f(x) \) and \( g(x) \) are actually defined at \( x = 3 \), their behavior or trend as they near this value can be characterized by the numbers 7 and 5, respectively.

• \( f(x) \) approaches 7 as \( x \) draws close to 3.
• \( g(x) \) veers towards 5 as \( x \) nears 3.

By understanding these behaviors, we can accurately combine these trends using rules like the Limit Addition Rule.

Calculus Problem Solving

Problem solving in calculus often involves applying the rules and properties of limits effectively. To determine the truth of a statement concerning the limits of functions, we rely on proven mathematical principles like the Limit Addition Rule. Breaking down a problem into manageable steps can simplify the process and ensure accuracy.
To solve a calculus problem efficiently:

• Begin by diligently understanding the given information and known limits.
• Apply an appropriate rule or theorem to combine or manipulate these limits.
• Calculate the result and compare it to any given statements or hypotheses.
• Evaluate whether the statement holds true or provide a counterexample if it does not.

In this exercise, each step followed the above plan, confirming the truth of the provided limits by using logical reasoning grounded on foundational calculus rules. Familiarity with these strategies will make calculus problem solving more intuitive and less daunting.