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Understanding vertical asymptotes is crucial when dealing with rational functions like the one given in the exercise. A vertical asymptote is a vertical line that the graph of a function approaches but never actually touches or crosses. This occurs at values of x that make the denominator zero, causing the function to be undefined at those points.

For the function \( f(x) = \frac{x^2 + 4}{x - 1} \), setting the denominator equal to zero will help find the vertical asymptote. Here, solving \( x - 1 = 0 \) results in \( x = 1 \). Therefore, \( x = 1 \) is where this particular function has a vertical asymptote.

These lines can help map out the graph since they indicate behavior changes where the function isn't defined. In practice, it's important to check the denominator in any rational function to effectively identify vertical asymptotes.

Horizontal asymptotes provide insight into the end behavior of a rational function, showing where the function tends to go as x approaches infinity (positive or negative). Not every function will have a horizontal asymptote, but when they do, they're valuable for indicating long-term behavior.

To determine if a function, such as \( f(x) = \frac{x^2 + 4}{x - 1} \), has a horizontal asymptote, you compare the degrees of the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, divide the leading coefficients of the numerator and denominator for the horizontal asymptote.
• If the numerator's degree is greater, as with this function, there is no horizontal asymptote.

The absence of a horizontal asymptote here points toward a different asymptotic behavior, which leads us to consider oblique asymptotes.

When the degree of the numerator is exactly one more than that of the denominator, an oblique (or slant) asymptote appears. This scenario happens because, as you divide the polynomial, the quotient takes a linear form. This asymptote represents the line that the graph of the function will approach at extreme values of x.

For the exercise function \( f(x) = \frac{x^2 + 4}{x - 1} \), since the numerator's degree (2) is one more than the denominator's degree (1), we have an oblique asymptote. We find this by performing polynomial division of \( x^2 + 4 \) by \( x - 1 \), resulting in the quotient \( x + 1 \).

Therefore, the oblique asymptote is \( y = x + 1 \). This linear equation provides a guide for the function's behavior as x moves toward positive or negative infinity, diverging from a horizontal line in contrast to how horizontal asymptotes act.