# Fundamental Mathematics/Calculus/Limit

## Limit

${\displaystyle \lim _{x\to a}f(x)}$

Finite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle c}$ if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently close (and not equal) to ${\displaystyle c}$ .

When this holds we write

${\displaystyle \lim _{x\to c}f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to c}$

Infinite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently large.

When this holds we write

${\displaystyle \lim _{x\to \infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to \infty }$

Similarly, we call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches negative infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently negative.

When this holds we write

${\displaystyle \lim _{x\to -\infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to -\infty }$