Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 41-48, find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a}[f(x)-g(x)]\)

Step-by-Step Solution

Answer

The short answer to the given question is: \(\lim _{x \rightarrow a}[f(x) - g(x)] = -1\)

Step 1: Determine the given limits

We are given: \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\)

Step 2: Apply limit properties

According to the limit properties, if \(\lim _{x \rightarrow a} f(x)=L_1\) and \(\lim _{x \rightarrow a} g(x)=L_2\), then \(\lim _{x \rightarrow a} [f(x) \pm g(x)] = L_1 \pm L_2\) We are supposed to find \(\lim _{x \rightarrow a}[f(x)-g(x)]\), so let's apply the properties: \(\lim _{x \rightarrow a}[f(x) - g(x)] = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x)\)

Step 3: Substitute the given limits

Since we already know the limits of the individual functions, we can substitute them into the equation: \(\lim _{x \rightarrow a}[f(x) - g(x)] = 3 - 4\)

Step 4: Calculate the final answer

Finally, we can calculate: \(\lim _{x \rightarrow a}[f(x) - g(x)] = -1\) So, the limit of \([f(x) - g(x)]\) as \(x\) approaches \(a\) is \(-1\).

Key Concepts

Limit Properties

Limits are essential in calculus for analyzing the behavior of functions as their input approaches a certain value. **Limit properties** are crucial because they allow us to evaluate complex limits by breaking them down into simpler parts. Here are some vital limit properties to understand:

• If you add or subtract two functions, the limit of their sum or difference is the sum or difference of their limits. For example, if \( \lim\limits_{x \to a} f(x) = L_1 \) and \( \lim\limits_{x \to a} g(x) = L_2 \), then \( \lim\limits_{x \to a} [f(x) + g(x)] = L_1 + L_2 \) and \( \lim\limits_{x \to a} [f(x) - g(x)] = L_1 - L_2 \).


• The limit of a constant is the constant itself. So, \( \lim\limits_{x \to a} c = c \).


• When multiplying functions, the limit of the product is the product of the limits. Thus, \( \lim\limits_{x \to a} [f(x) \cdot g(x)] = L_1 \cdot L_2 \).

Understanding these properties helps to simplify the calculation of limits, making it easier to solve problems like the one given in the exercise.

Continuity

Continuity is a property of functions that tells us if the function behaves nicely and predictably without any jumps or breaks. A function is continuous at a specific point if you can draw the function at that point without lifting your pencil. Here's what you need to understand about continuity:

• A function \( f(x) \) is continuous at a point \( x = a \) if \( \lim\limits_{x \to a} f(x) = f(a) \). This means the limit of the function as \( x \) approaches \( a \) should be equal to the function's value at \( a \).


• For a function to be continuous over an interval, it must be continuous at every point within that interval.


• Discontinuities can occur when there are jumps, holes, or vertical asymptotes in the graph of the function.

Understanding continuity ensures that you know where a function can be expected to behave predictably.

Function Operations

Functions can be combined and manipulated in various ways, which is an essential part of calculus. Here are some basic operations you can perform on functions and how they relate to limits:

• **Addition and Subtraction**: As seen in the exercise, when you add or subtract two functions \( f(x) \) and \( g(x) \), the result can be written as \( [f(x) + g(x)] \) or \( [f(x) - g(x)] \). You calculate their limits by using their individual limits.


• **Multiplication**: When multiplying functions, use the property of the product of limits. For \( [f(x) \cdot g(x)] \), the limit is \( \lim\limits_{x \to a} f(x) \cdot \lim\limits_{x \to a} g(x) \).


• **Division**: When dividing two functions, it's similar to multiplication. The limit is \( \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)} \), assuming that \( g(x) \) is not approaching zero.

By understanding how to perform these operations, you can tackle more advanced calculus problems effectively. Function operations are foundational for working with limits and derivatives.