Problem 49 - Calculus and its applications

Question

Is the function given by \(g(x)=x^{2}-3 x\) continuous at \(x=4 ?\) Why or why not?

Step-by-Step Solution

Answer

The function \( g(x) = x^2 - 3x \) is continuous at \( x=4 \) because the limit equals the function value at this point.

Step 1: Understanding the Continuity

A function is continuous at a point if it satisfies three conditions: 1) the function is defined at that point, 2) the limit as the variable approaches that point exists, and 3) the value of the function at that point is equal to the limit as the variable approaches that point.

Step 1: Checking the Function Value

Calculate the value of the function at the point of interest: \[ g(4) = 4^2 - 3 \times 4 = 16 - 12 = 4 \]. So, the function is defined at \( x=4 \) and \( g(4) = 4 \).

Step 2: Finding the Limit as x Approaches 4

To find the limit as \( x \) approaches 4, evaluate the limit of \( g(x) \): \[ \lim_{{x \to 4}} (x^2 - 3x) = \lim_{{x \to 4}} (x^2) - \lim_{{x \to 4}} (3x) \]. Calculate each: \[ \lim_{{x \to 4}} (x^2) = 16 \] and \[ \lim_{{x \to 4}} (3x) = 12 \]. Thus, \[ \lim_{{x \to 4}} (x^2 - 3x) = 16 - 12 = 4 \].

Step 3: Comparing the Function Value and the Limit

The value of the function at \( x=4 \) is 4, and the limit as \( x \) approaches 4 is also 4. Since these are equal, all the conditions for continuity are met at \( x=4 \).

Key Concepts

Limits

Limits are a fundamental concept in calculus used to understand how a function behaves as it approaches a certain point. Think of the limit as the value that a function approaches as the input gets closer and closer to a specific point. For the function \( g(x) = x^2 - 3x \), when we talk about the limit as \( x \) approaches 4, we mean what value does \( g(x) \) get closer to as \( x \) gets closer to 4.
Calculating the limit involves direct evaluation for many functions, especially polynomials. This means you simply substitute the value into the function to find what it approaches. For \( g(x) \), we calculate the limit as:

• \( \lim_{{x \to 4}} (x^2) = 16 \)
• \( \lim_{{x \to 4}} (3x) = 12 \)

Combining these, the limit is \( \lim_{{x \to 4}} (x^2 - 3x) = 4 \). Thus, as \( x \) nears 4, \( g(x) \) approaches the value 4.
Limits are crucial for defining when a function is continuous and in many other calculus concepts like derivatives and integrals.

Evaluating Functions

Evaluating a function simply means calculating its value at a particular point. For continuity, this step is vital because it involves checking the actual value of the function at a specified point.
To evaluate the function \( g(x) = x^2 - 3x \) at \( x = 4 \), we substitute 4 into the function:

• \( g(4) = 4^2 - 3 \times 4 = 16 - 12 = 4 \)

The result shows that at \( x = 4 \), the function \( g(x) \) has a value of 4.
This means that the function is defined at \( x = 4 \) and provides the exact value, which is crucial when determining function continuity. In essence, evaluating a function ensures we understand its behavior specifically at key points.

Polynomial Functions

Polynomial functions are expressions involving sums of powers of variables, each multiplied by coefficients. They are among the simplest and most familiar types of functions in algebra and calculus.
For example, the function \( g(x) = x^2 - 3x \) is a polynomial. It is specifically a quadratic polynomial, which means its highest degree term is \( x^2 \).
Here’s why polynomial functions matter:

• They are continuous everywhere, which makes them easy to work with for both limits and derivatives.
• They can be used to approximate other functions and are the foundation for more complex mathematical models.

Because polynomials, like \( g(x) \), are defined for all real numbers, calculating their limits or function values is straightforward, often simply involving substitution. They behave predictably, making them a sturdy tool in a mathematician’s toolkit. Understanding polynomials lays the groundwork for tackling more advanced calculus problems.