Mathematics

Asymptotes

Calculation and analysis of vertical, horizontal, and oblique (slant) asymptotes, as well as parabolic branches in rational functions.

Vertical Asymptotes: We analyze the one-sided limits at the points of discontinuity, where the function is undefined, and at the endpoints of the domain.
lim⁡x→a−f(x)=±∞andlim⁡x→a+f(x)=±∞\lim_{x\to a^-} f(x) = \pm\infty \quad \text{and} \quad \lim_{x\to a^+} f(x) = \pm\infty
The vertical asymptote has the equation x=ax = a
Example:
f(x)=2xx−2  ⟹  Dom=R−{2}f(x) = \frac{2x}{x-2} \implies Dom = \mathbb{R} - \{2\}
lim⁡x→2−2xx−2=−∞\displaystyle \lim_{x\to 2^-} \frac{2x}{x-2} = -\infty
lim⁡x→2+2xx−2=+∞\displaystyle \lim_{x\to 2^+} \frac{2x}{x-2} = +\infty
Asymptote: x=2x=2
Graph of a function with a vertical asymptote at x=2
Horizontal Asymptotes: We will have horizontal asymptotes when the limits at infinity exist:
lim⁡x→±∞f(x)=k\lim_{x\to \pm\infty} f(x) = k
In polynomial rational functions, there will be a horizontal asymptote if the degree of the numerator is equal to or less than the degree of the denominator.

The horizontal asymptote has the equation y=ky = k
Example:
f(x)=2xx−2\displaystyle f(x) = \frac{2x}{x-2}
lim⁡x→+∞2xx−2=2\displaystyle \lim_{x\to +\infty} \frac{2x}{x-2} = 2
lim⁡x→−∞2xx−2=2\displaystyle \lim_{x\to -\infty} \frac{2x}{x-2} = 2
Asymptote: y=2y=2
Graph of a function with a horizontal asymptote at y=2
Oblique Asymptotes: We will have oblique asymptotes when the following limit is finite:
lim⁡x→±∞f(x)x=m\lim_{x\to \pm\infty} \frac{f(x)}{x} = m
In polynomial rational functions, there will be an oblique asymptote if the degree of the numerator is exactly one more than the degree of the denominator.

It has the equation of a line: y=mx+ny = mx + n, where:
m=lim⁡x→±∞f(x)xm = \lim_{x\to \pm\infty} \frac{f(x)}{x}
n=lim⁡x→±∞(f(x)−mx)n = \lim_{x\to \pm\infty} (f(x) - mx)
Example:
f(x)=x2−xx−3\displaystyle f(x) = \frac{x^2 - x}{x - 3}
m=lim⁡x→±∞f(x)x=lim⁡x→±∞x2−xx2−3x=1\displaystyle m = \lim_{x\to \pm\infty} \frac{f(x)}{x} = \lim_{x\to \pm\infty} \frac{x^2 - x}{x^2 - 3x} = 1
n=lim⁡x→±∞(f(x)−x)=lim⁡x→±∞x2−x−x2+3xx−3=2\displaystyle n = \lim_{x\to \pm\infty} (f(x) - x) = \lim_{x\to \pm\infty} \frac{x^2 - x - x^2 + 3x}{x - 3} = 2
Asymptote: y=x+2y = x + 2
Graph of a function with an oblique asymptote at y=x+2
Horizontal and oblique asymptotes are mutually exclusive; if one type exists, the other will not.
Parabolic branches: We will have parabolic branches when:
lim⁡x→±∞f(x)x=±∞\lim_{x\to \pm\infty} \frac{f(x)}{x} = \pm\infty
Example:
f(x)=x3−1\displaystyle f(x) = x^3 - 1
lim⁡x→±∞f(x)x=lim⁡x→±∞x3−1x=∞\displaystyle \lim_{x\to \pm\infty} \frac{f(x)}{x} = \lim_{x\to \pm\infty} \frac{x^3 - 1}{x} = \infty
Graph of a function with a parabolic branch
For a polynomial rational function, it is easy to determine the types of asymptotes it will have by simply inspecting the domain and the degrees of the polynomials in the numerator and denominator.

On one hand, the points where the function is undefined, or the points that make the denominator zero, are points of vertical asymptotes.

On the other hand, horizontal and oblique asymptotes follow these criteria:
  • If degree of numerator < degree of denominator, there is a horizontal asymptote at y=0y=0
  • If degree of numerator = degree of denominator, there is a horizontal asymptote at y=ky=k, which is determined by the limit at infinity as seen before:
    lim⁡x→±∞f(x)=k\displaystyle \lim_{x\to \pm\infty} f(x) = k
  • If degree of numerator = degree of denominator + 1, there is an oblique asymptote
  • If degree of numerator > degree of denominator + 1, there are no horizontal or oblique asymptotes. There will be a parabolic branch.