Problem 23 - Calculus: Early Transcendentals

Question

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right)$$

Step-by-Step Solution

Answer

Answer: The limit as $$x$$ approaches negative infinity is $$+\infty$$.

Step 1: Identify the dominating term

In this polynomial, the dominating term is $$-3x^{16}$$, because it has the highest power of x (16). The other term, 2, will have negligible impact as x approaches negative infinity. #----#

Step 2: Determine the limit

We will now determine the limit as x approaches negative infinity. Because the dominating term is $$-3x^{16}$$, the function's behavior will be primarily determined by this term as x approaches negative infinity. Since the power of x is even (16), the function will tend towards positive infinity if x is large and negative. Therefore, the limit as x approaches negative infinity is: $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right) = \lim _{x \rightarrow-\infty}\left(-3 x^{16}\right) = +\infty$$

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve terms with variables raised to whole number powers and constant coefficients. They can take various forms, such as linear, quadratic, cubic, and higher-order polynomials. A typical polynomial function may look like:

• \(a_nx^n + a_{n-1}x^{n-1} + \, ext{...} \, + a_1x + a_0\)

Here, the \(a\)s are coefficients, and \(n\) is a non-negative integer representing the highest degree in the polynomial.
Polynomial functions are smooth and continuous, meaning that you can draw their graphs without lifting the pencil from the paper. These functions are also closed under addition, subtraction, multiplication, and other operations, making them an essential class of functions in calculus and algebra.
When working with polynomials, each term is a product of a constant and a power of the variable. The highest power term often plays a significant role in understanding the function's behavior, especially when assessing limits.

Dominating Term

The dominating term in a polynomial is the term with the highest power of the variable, often referred to as the leading term. It greatly influences the polynomial's behavior, particularly as the variable grows indefinitely in magnitude either positively or negatively.
For example, in the polynomial

• \(-3x^{16} + 2\)

the term \(-3x^{16}\) dominates because it features \(x\) raised to the 16th power, which is larger than any other powers present. This dominance means that as \(x\) increases or decreases significantly, the overall behavior of the polynomial closely follows this high-power term.
The effect of the dominating term becomes more pronounced while analyzing limits, especially in the context of behavior at infinity. This distinction helps in determining whether a polynomial approaches infinity, negative infinity, or zero as \(x\) approaches extremely large or small values.

Behavior at Infinity

Understanding the behavior of functions as they approach infinity is crucial for analyzing limits. For polynomial functions, this generally means focusing on what happens to the function's value as the variable \(x\) becomes very large or very small.
When we discuss a limit approaching infinity, we address how the function grows as the input grows larger (positive infinity) or smaller (negative infinity).

• If the highest power of \(x\) is even, the function will either grow towards positive infinity or negative infinity, depending on the sign of the coefficient.
• When examining the polynomial \(-3x^{16} + 2\), the even power of 16 ensures the terms will tend towards the same infinities as \(x\) becomes increasingly negative.

Since the highest power is even, and the coefficient is negative, the function behaves oppositely when \(x\) approaches negative infinity, resulting in the polynomial itself tending towards positive infinity. Understanding this concept helps predict the limit result, which in this exercise was \(+\infty\).