Problem 45 - Precalculus

Question

Evaluate the limits using the limit properties. $$\lim _{x \rightarrow-4}\left(x^{3}-5\right)$$

Step-by-Step Solution

Answer

The limit is -69.

Step 1: Analyze the Function

The function given is a polynomial, specifically, it is \( f(x) = x^3 - 5 \). Polynomials are continuous functions, which means we can evaluate their limits by direct substitution.

Step 2: Apply Limit Properties

According to the limit properties, the limit of a polynomial function at a point can be found by directly substituting the point into the function. Thus, to calculate the limit as \( x \to -4 \), substitute \( -4 \) into \( x^3 - 5 \).

Step 3: Substitute \( x = -4 \) into the Function

Replace \( x \) in the function with \( -4 \):\[(-4)^3 - 5\]

Step 4: Calculate the Expression

First, calculate \( (-4)^3 \):\[ (-4) imes (-4) imes (-4) = -64 \]Then substitute back into the expression:\[ -64 - 5 = -69 \]

Step 5: Conclusion

Therefore, the limit of \( f(x) = x^3 - 5 \) as \( x \to -4 \) is \( -69 \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients. These functions only involve operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A general form of a polynomial function is given as:

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)

Properties of Polynomial Functions

• Degree: The degree of a polynomial is the highest power of the variable in the expression. For example, in our function \( f(x) = x^3 - 5 \), the degree is 3.
• Continuous: Polynomial functions are continuous for all values of \( x \). This means there are no breaks, holes, or jumps in the graph. One can smoothly trace these functions without lifting the pencil off the paper.
• Smoothness: The graph of a polynomial function is always smooth and rounded, not having any sharp corners or cusps.

Polynomials play a crucial role in calculus as they frequently appear in real-world applications and mathematical problems. They provide a rich context for exploring various calculus concepts like limits, derivatives, and integrals.

Direct Substitution

Direct substitution is a method used to evaluate limits, especially when dealing with polynomial functions or other continuous functions. This technique involves directly replacing the variable with the value it approaches.

How Direct Substitution Works

When you have a polynomial function and you need to evaluate its limit as \( x \) approaches a certain value, direct substitution allows you to:

• Simply substitute the given value for \( x \) into the function.
• Compute the result of the expression obtained after substitution.

In our example, the limit of the polynomial function \( f(x) = x^3 - 5 \) as \( x \to -4 \) can be found by directly replacing \( x \) with \(-4\). This straightforward method is efficient because polynomials are continuous everywhere in their domain.

Why Use Direct Substitution?

• It's simple and requires minimal steps, making it quick to perform.
• For continuous functions, it guarantees that the calculated value is indeed the actual limit.

When a function is continuous and defined at the point approaching, direct substitution is almost always applicable and reliable.

Continuity of Functions

The concept of continuity is fundamental in calculus and helps us understand behaviors of functions. A function is continuous at a point if the following conditions are satisfied:

• The function is defined at that point.
• The limit of the function as \( x \) approaches that point exists.
• The value of the function at that point equals the limit of the function as \( x \) approaches that same point.

Polynomials and Continuity

Polynomials are perhaps the simplest class of functions when it comes to continuity. They are continuous for every real number, which means they have no interruptions at any point across the real number line. This attribute makes them very friendly for calculating limits. If you are tasked with finding the limit of a polynomial function as \( x \) approaches a particular value, you can be confident that:

• The function will be smooth and uninterrupted at that point.
• Direct substitution will give you the exact limit you are seeking.

Understanding the continuity of functions, especially in the context of polynomials, provides a solid foundation for approaching more complex calculus problems involving limits, derivatives, and integrals.