Asymptotes of Real Functions

Asymptotes

Definition: Vertical Asymptote

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The line $x=c\in \mathbb{R}$ is a vertical asymptote of $f$ if $f$ has at least one infinite one-sided limit at $c$, i.e. at least one of the following holds:

• $\underset{x\to c^{-}}{\lim }f(x)=-\infty$

• $\underset{x\to c^{+}}{\lim }f(x)=-\infty$

• $\underset{x\to c^{-}}{\lim }f(x)=\infty$

• $\underset{x\to c^{+}}{\lim }f(x)=\infty$

INTUITION

Intuitively, this definition means that the value of $f(x)$ gets closer and closer to $\pm \infty$ as $x$ approaches $c$ either from the left or from the right.

Definition: Horizontal Asymptote

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The line $y=a\in \mathbb{R}$ is a horizontal asymptote of $f$ if the limit of $f$ as $x$ approaches positive or negative infinity is $a$, i.e. if at least one of the following holds:

• $\underset{x\to -\infty }{\lim }f(x)=a$

• $\underset{x\to \infty }{\lim }f(x)=a$

INTUITION

Intuitively, this definition means that the value of $f(x)$ gets closer and closer to $a$ as $x$ approaches either positive or negative infinity.

Definition: Oblique Asymptote

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The line $y=ax+b$ is an oblique or slanted asymptote of $f$ if at least one of the following is true:

• $\underset{x\to -\infty }{\lim }[f(x)-(ax+b)]=0$

• $\underset{x\to \infty }{\lim }[f(x)-(ax+b)]=0$

INTUITION

Intuitively, this definition means $f(x)$ gets closer and closer to the line $y=ax+b$ as $x$ approaches either positive or negative infinity.

Properties

Theorem: Oblique Asymptotes

Let $f$ be a real function.

The straight line $y=ax+b$ is an asymptote of $f$ if and only if the limits ${\lim }_{x\to \pm \infty }\frac{f(x)}{x}$ and ${\lim }_{x\to \pm \infty }(f(x)-ax)$ exist and

$$\begin{array}{rl}a& =\underset{x\to \pm \infty }{\lim }\frac{f(x)}{x}\\ & \\ b& =\underset{x\to \pm \infty }{\lim }(f(x)-ax)\end{array}$$

PROOF

TODO