Vertical and Oblique Asymptotes of Rational Functions

Limits and Asymptotic Behavior of Functions

Understanding Limits Near a Point

In precalculus, it is important to describe the behavior of a function f as x approaches a value a, but not necessarily at x = a. This is done using the concept of limits.

• Limit from both sides: $\lim\limits_{x \to a} f(x) = L$ means as x approaches a from both sides, f(x) approaches L.

• Left-hand limit: $\lim\limits_{x \to a^-} f(x) = L$ means x approaches a from the left.

• Right-hand limit: $\lim\limits_{x \to a^+} f(x) = L$ means x approaches a from the right.

Vertical Asymptotes

A vertical asymptote of a function f(x) is a vertical line x = a where the function increases or decreases without bound as x approaches a from either side. This occurs if at least one of the following is true:

• $\lim\limits_{x \to a^-} f(x) = \pm \infty$

• $\lim\limits_{x \to a^+} f(x) = \pm \infty$

Example: $f(x) = \frac{1}{x}$ has a vertical asymptote at x = 0.

Rational Functions and Vertical Asymptotes

Definition and Identification

A rational function is in lowest terms if the numerator p(x) and denominator q(x) have no common factors. A rational function in lowest terms will have a vertical asymptote at x = r if r is a real zero of the denominator.

• Example: $p(x) = \frac{2(x^2-2)}{x^2-5}$ has vertical asymptotes at $x = \sqrt{5}$ and $x = -\sqrt{5}$.

• No vertical asymptotes: If the denominator has no real zeros, there are no vertical asymptotes.

• Removable discontinuity (hole): If a factor cancels in numerator and denominator, the function has a hole, not a vertical asymptote.

Function	Vertical Asymptotes	Notes
$a(x) = \frac{1}{x^2-3}$	$x = \sqrt{3},\ -\sqrt{3}$	Denominator zeros
$b(x) = \frac{2}{x^2+2}$	None	No real zeros in denominator
$c(x) = \frac{(x+1)(x-2)}{x-1}$	$x = 1$	Vertical asymptote at $x=1$
$d(x) = \frac{x^2-1}{x-1}$	None	Hole at $x=1$ (removable)

Note: $c(x) = \frac{x^2-1}{x-1}$ is not the same function as $d(x) = x+1$; they have different domains.

Oblique (Slant) Asymptotes

Definition and Identification

If a rational function $R(x)$ approaches a linear expression $mx+b$ as $x \to \infty$ or $x \to -\infty$, then the line $y = mx + b$ is called an oblique (slant) asymptote. This occurs when the degree of the numerator is exactly one more than the degree of the denominator.

• Example: $H(x) = \frac{3x^2 - x}{x}$

• Divide numerator by denominator: $\frac{3x^2 - x}{x} = 3x - 1$

• As $x \to \infty$, $H(x) \to 3x - 1$

• Thus, $y = 3x - 1$ is the slant asymptote of $H$.

Summary Table:

Type of Asymptote	How to Find	Example
Vertical	Set denominator = 0 (after simplification)	$f(x) = \frac{1}{x}$ at $x=0$
Oblique (Slant)	Degree numerator = 1 + degree denominator; divide	$H(x) = \frac{3x^2-x}{x}$, $y=3x-1$