Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-15\), state whether the indicated function is continuous at \(3 .\) If it is not contimuous, tell why. $$ f(x)=(x-3)(x-4) \quad \text { 2. } g(x)=x^{2}-9 $$

Step-by-Step Solution

Answer

The function \( g(x) = x^2 - 9 \) is continuous at \( x = 3 \).

Step 1: Understand the Continuity Definition

A function is continuous at a point if the following three conditions are satisfied: 1. The function is defined at that point. 2. The limit of the function exists at that point. 3. The limit of the function, as it approaches the point, is equal to the function's value at that point.

Step 2: Check if g(x) is Defined at x = 3

The function given is \( g(x) = x^2 - 9 \). Evaluate \( g(3) \) to see if the function is defined at \( x = 3 \): \( g(3) = 3^2 - 9 = 9 - 9 = 0 \). Therefore, \( g(x) \) is defined at \( x = 3 \).

Step 3: Find the Limit of g(x) as x Approaches 3

Calculate the limit of \( g(x) \) as \( x \) approaches 3. Since \( g(x) = x^2 - 9 \) is a polynomial, limits of polynomials exist and can be found by direct substitution.Thus, \( \lim_{{x \to 3}} g(x) = 3^2 - 9 = 0 \).

Step 4: Compare the Function Value with the Limit

Compare \( g(3) = 0 \) with \( \lim_{{x \to 3}} g(x) = 0 \). Since they are equal, the function \( g(x) = x^2 - 9 \) meets the condition of continuity at \( x = 3 \).

Step 5: Conclusion

Since all three conditions for continuity are satisfied (function defined at the point, limit exists, and limit equals the function's value), the function \( g(x) \) is continuous at \( x = 3 \).

Key Concepts

Functions

In mathematics, a **function** is a relation between a set of inputs and a set of permissible outputs. Each input is related to exactly one output. It can be visualized as a machine that takes an input, processes it, and gives an output. Here's what you should know about functions:

• Functions can be represented in different forms, such as equations, graphs, and tables.
• The notation \( f(x) \) is commonly used to denote a function named \( f \) with \( x \) as the input.
• Functions can describe real-world situations, like calculating areas, predicting populations, or measuring speed and time.

A fundamental property of functions is **continuity**, especially when dealing with calculus. Understanding the behavior of functions at specific points helps in understanding the broader behavior of graphs and real-world models. Ensuring a function is continuous at a point means looking at its defined value, the limit as it approaches, and ensuring both agree and exist.

Polynomials

**Polynomials** are a special type of function characterized by the sum of terms, each including a variable raised to an integer power, and multiplied by coefficients. For example, the function \( g(x) = x^2 - 9 \) from the exercise is a polynomial. Here's why polynomials are essential:

• Polynomials are easy to differentiate and integrate, making them fundamental in calculus.
• They allow us to model simple curves and approximate complex ones.
• The degree of a polynomial determines its behavior and shape. A quadratic, for example, has a parabolic shape.

The properties of polynomials, such as their continuity, can often be easily checked by substituting a value into the polynomial. Because polynomials are continuous everywhere on their domains — which is all real numbers — checking their behavior at any point is straightforward.

Limits in Calculus

**Limits in calculus** are crucial for understanding the behavior of functions as their input approaches a certain value. In the exercise given, understanding limits helps determine continuity at a specific point.

• The limit describes the value that a function approaches as the input gets closer to a specific number.
• Limits help define derivatives (rates of change) and integrals (areas beneath curves).
• The concept of a limit is used to define, in precise terms, what we mean by a point being close to another point.

To find a limit for polynomials, we often use **direct substitution**. This means plugging the approaching input value directly into the polynomial, since polynomials are smooth and continuous. For instance, in the context of the provided exercise, evaluating \( \lim_{{x \to 3}} g(x) = 3^2 - 9 \), we simply substituted \( 3 \) directly into \( g(x) \). This helps affirm the continuity of polynomial functions at any point.