Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=3/(x-5)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{3}{x-5}\).
2. Find the vertical asymptote(s) by setting the denominator equal to zero and solving for \(x\): \(x - 5 = 0\) which gives \(x = 5\). This means there is a vertical asymptote at \(x = 5\).
3. Determine the horizontal or oblique asymptote by analyzing the degrees of the numerator and denominator. The numerator is a constant (degree 0), and the denominator is degree 1.
4. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
5. Conclude that there are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. For ƒ(x) = 3/(x-5), setting the denominator x-5 = 0 gives x = 5, indicating a vertical asymptote at x = 5.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, compare the degrees of numerator and denominator: if the numerator's degree is less, the horizontal asymptote is y = 0; if equal, it is the ratio of leading coefficients.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the denominator's degree. They are found by performing polynomial division. For ƒ(x) = 3/(x-5), since the numerator degree is less, no oblique asymptote exists.