Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 2}\left(x^{3}+4 x^{2}-5\right)$$

Step-by-Step Solution

Answer

The limit is 19.

Step 1: Identify the Form of the Function

The function provided is a polynomial: \( f(x) = x^3 + 4x^2 - 5 \). Polynomials are continuous everywhere on \( \mathbb{R} \). Therefore, we can find the limit by direct substitution.

Step 2: Substitute \( x = 2 \) into the Function

Substitute \( x = 2 \) into the polynomial to find the limit. This means evaluating \( f(2) = 2^3 + 4 \times 2^2 - 5 \).

Step 3: Calculate the Expression

Calculate each part of the polynomial expression:- \( 2^3 = 8 \)- \( 4 \times 2^2 = 4 \times 4 = 16 \)- Combine these with the constant: \( 8 + 16 - 5 \).

Step 4: Perform the Final Calculation

Add the calculated expressions: \( 8 + 16 - 5 = 19 \). This is the value of the limit.

Key Concepts

Understanding Polynomials

Polynomials are fundamental mathematical expressions that involve variables and coefficients. They consist of terms in the form of \( ax^n \), where \( a \) represents a coefficient, \( x \) is a variable, and \( n \) is a non-negative integer called the degree of the term. In simpler terms, a polynomial is a sum of these terms put together.

The polynomial given in the exercise is \( f(x) = x^3 + 4x^2 - 5 \). This specific polynomial has three terms:

• \( x^3 \), with a coefficient of 1.
• \( 4x^2 \), with a coefficient of 4.
• -5, which is a constant term.

Understanding polynomials is crucial as they frequently appear in various areas of mathematics, modeling real-world situations due to their simplicity and versatility. They can be easily added, subtracted, multiplied, and divided, making them a cornerstone in algebra.

Exploring Continuous Functions

Continuous functions are an essential concept in calculus, and understanding them helps in dealing with limits effectively. The idea of continuity intuitively means that the graph of the function can be drawn without lifting the pencil from the paper. Specifically, a function is said to be continuous at a point if three conditions are satisfied:

• The function is defined at that point.
• The limit of the function exists as it approaches the point.
• The limit equals the function's value at that point.

Polynomials, like the one in our exercise \( f(x) = x^3 + 4x^2 - 5 \), are an example of continuous functions. They have no breaks, jumps, or holes in their graphs, which means they are continuous everywhere on the real number line. This property allows us to calculate limits using direct substitution rather easily.

Applying Direct Substitution

Direct substitution is one of the simplest methods to evaluate limits, especially when dealing with continuous functions like polynomials. The process involves plugging the value that \( x \) approaches directly into the function. In our exercise, we wish to find \( \lim_{x \to 2} \) for the function \( f(x) = x^3 + 4x^2 - 5 \). Since the polynomial is continuous, we can perform direct substitution, evaluating \( f(2) \). Here's how it works:

• Substitute \( x = 2 \) into \( f(x) \), resulting in \( 2^3 + 4 \times 2^2 - 5 \).
• Calculate each term: \( 2^3 = 8 \), \( 4 \times 4 = 16 \), and then combine the results: \( 8 + 16 - 5 \).
• This gives us \( 19 \), which is the limit of the polynomial as \( x \) approaches 2.

This method highlights the ease of working with continuous functions, simplifying the limit evaluation process to simple arithmetic.