Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4.
$ƒ (x) = (x^{2} + 4) / (x - 1)$

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Identify the vertical asymptotes by finding the values of $x$ that make the denominator zero. Set the denominator equal to zero: $x - 1 = 0$ and solve for $x$.

Determine the vertical asymptote(s) from the solution(s) to the denominator equation. These are the values where the function is undefined and the graph may have vertical asymptotes.

Find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. The numerator is $x^2 + 4$ (degree 2) and the denominator is $x - 1$ (degree 1).

Since the degree of the numerator is greater than the degree of the denominator by exactly 1, perform polynomial long division of $x^2 + 4$ by $x - 1$ to find the oblique asymptote.

Write the equation of the oblique asymptote as the quotient obtained from the division (ignoring the remainder). This line represents the asymptote that the graph approaches as $x$ goes to infinity or negative infinity.

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Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x, excluding any values that also make the numerator zero (which may indicate holes instead).

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. For rational functions, compare the degrees of the numerator and denominator: if the numerator's degree is less, the asymptote is y=0; if equal, it's the ratio of leading coefficients; if greater, no horizontal asymptote exists.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They are found by performing polynomial long division of the numerator by the denominator; the quotient (without the remainder) gives the equation of the slant asymptote, representing the end behavior of the function.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4.ƒ(x)=(x2+4)/(x−1)ƒ(x)=(x^2+4)/(x-1)