The four fundamental cases

Depending on how $x$ tends (to a finite number or to infinity) and on how $f(x)$ tends, four fundamental cases are distinguished.

Notation	$x$ tends to	$f(x)$ tends to
$\lim_{x\to x_0} f(x) = L$	finite number	finite number
$\lim_{x\to x_0} f(x) = \pm\infty$	finite number	infinity (vertical asymptote)
$\lim_{x\to\pm\infty} f(x) = L$	infinity	finite number (horizontal asymptote)
$\lim_{x\to\pm\infty} f(x) = \pm\infty$	infinity	infinity

The graph of the function $f(x) = \dfrac{1}{x-2}+1$ shows two of these behaviours simultaneously: a vertical asymptote at $x=2$ and a horizontal asymptote at $y=1$.

As $x\to 2^-$ the function tends to $-\infty$ (vertical asymptote); as $x\to 2^+$ it tends to $+\infty$; as $x\to\pm\infty$ it tends to $1$ (horizontal asymptote).

Links

Topics: Limits
Concepts: Asymptote · Horizontal asymptote · Vertical asymptote · Infinity · Limit · Right and left limit
Skills: Interpreting a graph