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A horizontal asymptote is a horizontal line that a graph approaches as the input (or x-value) goes towards positive or negative infinity. In the context of rational functions, finding the horizontal asymptote involves checking the degrees of the function's numerator and denominator.
In this particular exercise, the degrees of both the numerator \(x^2 - 2\) and the denominator \(x^2 - 4\) are 2. When the degrees are equal, the horizontal asymptote can be determined by dividing the leading coefficients.

Here, the leading coefficient of both the numerator and denominator is 1. Thus, the horizontal asymptote is found as follows:

• Divide the leading coefficient of the numerator (1) by the leading coefficient of the denominator (1).
• This results in the equation \(y = \frac{1}{1} = 1\).

Therefore, the horizontal asymptote for this function is the line \(y = 1\). The graph will approach this line as x becomes very large or very small, but it will never actually touch it.

Vertical asymptotes are vertical lines that a function approaches but never crosses or touches. For rational functions, vertical asymptotes are found where the denominator is zero and the numerator is not zero.
In the exercise given, the denominator of the function is \(x^2 - 4\). To find the vertical asymptotes, we set this equation to zero:
\[x^2 - 4 = 0\]

Factoring this expression gives \((x - 2)(x + 2) = 0\). Solving for x gives two roots:

• \(x - 2 = 0\), leading to \(x = 2\)
• \(x + 2 = 0\), leading to \(x = -2\)

These values of x are where the vertical asymptotes occur. They are \(x = 2\) and \(x = -2\). The function will approach these lines but will not cross them, creating a clear division in the graph's structure at these points.

Rational functions are a special type of function formed by the ratio of two polynomial expressions. That means they look like \(rac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
Understanding rational functions is crucial because they have distinctive features like asymptotes due to their inherent structure. As a general rule:

• The degree of the numerator and denominator determines the horizontal asymptote.
• Values that make the denominator zero but not the numerator provide the vertical asymptotes.

These functions often have restricted domains, as the values that make the denominator zero are not valid inputs. In this problem, the rational function \(f(x) = \frac{x^2 - 2}{x^2 - 4}\) has these classic characteristics. Recognizing the numerator and denominator aids in determining asymptotic behavior, leading to a clearer graph and understanding of behavior near asymptotes.