Problem 11 - Calculus

Question

In Exercises \(10-15,\) give \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow+\infty} f(x)\). $$f(x)=5+21 x-2 x^{3}$$

Step-by-Step Solution

Answer

\(\lim_{x \to -\infty} f(x) = \infty\) and \(\lim_{x \to +\infty} f(x) = -\infty\).

Step 1: Understanding the Function

We are given the function \(f(x) = 5 + 21x - 2x^3\). We need to find the limits as \(x\) approaches both negative and positive infinity.

Step 2: Identifying the Dominant Term

The function is a polynomial, and when dealing with limits at infinity, the dominant term is the one with the highest power. In this function, the term \(-2x^3\) has the highest power of 3.

Step 3: Calculating the Limit as \(x \to -\infty\)

As \(x\) approaches negative infinity, the term \(-2x^3\) will dominate. Since \(2x^3\) is negative when \(x\) is negative, and it is also multiplied by \(-2\), the expression becomes positive as \(x\) becomes very large in the negative direction. Thus, \(\lim_{x \to -\infty} f(x) = \infty\).

Step 4: Calculating the Limit as \(x \to +\infty\)

As \(x\) approaches positive infinity, the term \(-2x^3\) will again dominate. As \(x\) is positive, \(2x^3\) is positive, and the negative sign in front makes the term negative. Thus, the limit is driven negatively. So, \(\lim_{x \to +\infty} f(x) = -\infty\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving the sum of powers of a variable multiplied by coefficients. Each power of the variable is called a term, and the number of terms defines the polynomial's degree. For instance, the polynomial function \(f(x) = 5 + 21x - 2x^3\) has three terms: a constant \(5\), a linear term \(21x\), and a cubic term \(-2x^3\). The degree of a polynomial is the highest exponent of the variable, which, in this case, is 3 due to the \(-2x^3\) term.

Characteristics of polynomial functions include:

• They are smooth and continuous, without any breaks or holes.
• Higher-degree polynomials can have more complex curves with several turning points.
• Polynomials can be classified into different types based on their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.

Understanding polynomial functions is fundamental when exploring their behavior as they approach infinity.

Dominant Term

When analyzing polynomial functions, particularly when determining limits at infinity, the dominant term plays a crucial role. This term dictates the polynomial's behavior as the variable approaches very large positive or negative values. The dominant term is typically the one with the highest degree since it grows or decreases much faster than the other terms.

In the function \(f(x) = 5 + 21x - 2x^3\), the dominant term is \(-2x^3\). As \(x\) becomes extremely large or small, this term outweighs the others in magnitude. For instance, as \(x\) grows towards infinity, the \(-2x^3\) term becomes significantly larger compared to \(5\) or \(21x\), hence determining the overall behavior of the function.

Recognizing the dominant term simplifies the analysis and helps predict the function's end behavior, as the lower-degree terms become negligible in comparison.

Asymptotic Behavior

Asymptotic behavior in terms of polynomial functions involves analyzing how the function behaves as the variable \(x\) moves towards infinity or negative infinity. For polynomials like \(f(x) = 5 + 21x - 2x^3\), this is primarily governed by the dominant term.

When \(x\) approaches negative infinity, the term \(-2x^3\) takes precedence, and because of its negative sign, the cubed value of a negative number becomes positive. Thus, \(-2x^3\) drags the entire function towards positive infinity. Conversely, as \(x\) trends towards positive infinity, \(-2x^3\) becomes highly negative, leading the function towards negative infinity.

Hence, the asymptotic behavior provides insights into the function's long-term tendencies. Key takeaways include:

• The function's end-behavior is dictated by its dominant term.
• Tracking the sign and degree of this term helps predict whether the function approaches positive or negative infinity.

This understanding is crucial for graphical interpretation and solving real-world problems involving polynomial trends.