2.7 Asymptotes

1. Definition of Asymptotes

An asymptote describes the directional boundaries that a function progressively approaches but mathematically avoids at extreme bounds.

Vertical Asymptotes (VA): Evaluated at specific domain points $x=a$ where the functional output tends dynamically to $+\infty$ or $-\infty$. In simple fractional structures, this occurs exactly where the denominator equates to zero (providing the numerator is non-zero).

Horizontal Asymptotes (HA): Formed by determining the boundary output as $x$ tends to extremely large positive or negative ranges ($x \rightarrow \pm\infty$). The graph stabilizes approaching the horizontal threshold $y=b$.

EXAMPLE 1 (Evaluating Horizontal Asymptotes)

Consider the function $f(x) = \dfrac{4x+1}{2x-6}$. We can determine its horizontal asymptote using two analytical methods:

Method 1: Algebraic Manipulation
The function can be rewritten by splitting the numerator:

$$f(x) = \dfrac{4x + 1}{2x - 6} = \dfrac{2(2x - 6) + 13}{2x - 6} = \dfrac{2(2x - 6)}{2x - 6} + \dfrac{13}{2x - 6} = \mathbf{2 + \dfrac{13}{2x - 6}}$$

As $x$ tends to $+\infty$ or $-\infty$, the fraction $\dfrac{13}{2x - 6}$ approaches $0$.

Method 2: Dividing by the highest power of $x$
If we divide every term in the numerator and denominator by $x$, we obtain:

$$f(x) = \dfrac{\dfrac{4x}{x} + \dfrac{1}{x}}{\dfrac{2x}{x} - \dfrac{6}{x}} = \mathbf{\dfrac{4 + \dfrac{1}{x}}{2 - \dfrac{6}{x}}}$$

As $x$ tends to $+\infty$ or $-\infty$, the fractions $\dfrac{1}{x}$ and $\dfrac{6}{x}$ approach $0$.

In both cases $f(x)$, that is the value of $y$, approaches $2$. Therefore, the horizontal asymptote is $y = 2$. The vertical asymptote is found where the denominator is zero ($2x - 6 = 0 \Rightarrow \mathbf{x = 3}$).

Graph of $f(x) = \dfrac{4x+1}{2x-6}$

2. Rational Functions: $f(x) = \frac{Ax + B}{Cx + D}$

Functions possessing standard linear polynomials in the numerator and denominator reliably yield precisely one vertical and one horizontal asymptote.

Vertical Asymptote: Derived strictly by solving the denominator $Cx + D = 0 \Rightarrow x = -D/C$.
Horizontal Asymptote: Derived analytically by evaluating the ratio of leading coefficients: $y = \dfrac{A}{C}$.

EXAMPLE 2 (Rational Asymptote Derivations)

Function	Vertical Asymptote	Horizontal Asymptote
$f(x) = \dfrac{3x - 7}{x - 5}$	$x = 5$	$y = 3$
$f(x) = \dfrac{3x - 7}{2x - 5}$	$x = \dfrac{5}{2}$	$y = \dfrac{3}{2}$
$f(x) = \dfrac{8x - 7}{2x + 4}$	$x = -2$	$y = 4$
$f(x) = \dfrac{7}{x - 5}$	$x = 5$	$y = 0$ (coeff ratio $0/1$)
$f(x) = \dfrac{7}{x - 5} + 3$	$x = 5$	$y = 3$ (shifts HA up 3)

EXAMPLE 3 (Inverses)

Let $f(x) = \dfrac{3x+2}{x-4}$. The inverse evaluates precisely to $f^{-1}(x) = \dfrac{4x+2}{x-3}$. Notice the strict exchange of domain/range properties:

For $f(x)$ (blue): Domain restricts $x \ne 4$ (VA at $x=4$). HA evaluates to $y=3$ (Range restricts $y \ne 3$).
For $f^{-1}(x)$ (red): Domain restricts $x \ne 3$ (VA at $x=3$). HA evaluates to $y=4$ (Range restricts $y \ne 4$).

Self-Inverse Notice: A rational function where $f^{-1}(x) = f(x)$ creates symmetric asymptotes, e.g., $f(x) = \dfrac{2x+3}{x-2}$, where VA is $x=2$ and HA is $y=2$. The graph reflects perfectly onto itself across the line $y=x$.