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

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