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A polynomial function is a mathematical expression that can be composed of one or more terms, where each term contains a constant multiplied by a variable raised to a whole number power. These functions are expressed using the form:

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

where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer called the degree of the polynomial.

Polynomials are fundamental in algebra and calculus because their properties are different depending on the degree of the polynomial. The highest degree of the polynomial determines its most notable characteristics. When dealing with limits, particularly as \( x \) approaches infinity or negative infinity, only the leading term's degree and coefficient significantly impact the behavior.

In our exercise, the polynomials \( f(x) \) and \( g(x) \) simplify to their leading terms when assessing limits at infinity. This is because other terms become negligible as \( x \) grows larger in magnitude.

Even and odd functions have unique symmetry properties that influence the behavior of polynomial and other functions.

Even Functions have the property that

• \( f(-x) = f(x) \).

This symmetry means the graph of the function is mirrored across the y-axis. For polynomials, this happens when all non-zero power terms are even. An example could be \( x^2 \) or \( x^4 \). If \( f(x) \) is even, it behaves the same whether approaching positive or negative infinity.Odd Functions have the property that

• \( f(-x) = -f(x) \).

The graph of an odd function is symmetric about the origin, meaning rotating the graph 180 degrees around the origin does not change its appearance. Polynomial terms like \( x \) or \( x^3 \) are examples of odd-power terms. An odd \( n \) in the leading term of our polynomial would flip its sign when \( x \) moves from positive to negative infinity.

In our exercise, understanding whether the leading terms of \( f(x) \) or \( g(x) \) are even or odd helps predict the sign and value of their ratio's limit at negative infinity.

Rational functions involve ratios or quotients of polynomial functions. They take the form

• \( \frac{f(x)}{g(x)} \)

where both \( f(x) \) and \( g(x) \) are polynomials. Evaluating limits at infinity for rational functions focuses on the leading terms of the numerator and denominator, as other terms become negligible in magnitude.

In situations like the exercise given, understanding that the degree of the numerator equals the degree of the denominator is crucial. This balance ensures that the limit is neither 0 nor infinity, but a specific finite number, in our case, 2. This limit can differ at positive and negative infinity based on whether the degree of the polynomial is even or odd.

This specific understanding leads us to conclude in the exercise's solution that if the leading term's degree is even, the limit remains the same. However, if it is odd, the quotient's behavior flips in sign as \( x \) approaches negative infinity, leading to a limit of \(-2\) instead of \(2\). By recognizing these characteristics in rational functions, one can predict behavior at infinity more effectively.