Indeterminate form

In calculus, the expressions

${\displaystyle 0/0}$

${\displaystyle 0\cdot \infty }$

${\displaystyle 1^{\infty }}$

${\displaystyle 0^{0}}$

${\displaystyle \infty ^{0}}$

[others?]

are indeterminate forms; if f(x) and g(x) both approach 0 as x approaches some number, or x approaches ∞ or − ∞, then

${\displaystyle f(x)/g(x)}$

can approach any real number or ∞ or − ∞, or fail to converge to any point on the extended real line, depending on which functions f and g are; similar remarks are true of the other indeterminate forms displayed above. For example

${\displaystyle \lim _{x\rightarrow 0}{\sin(x) \over x}=1}$

and

${\displaystyle \lim _{x\rightarrow 49}{x-49 \over {\sqrt {x}}\,-7}=14.}$

In the first case, "0/0" becomes 1; in the second case, "0/0" becomes 14.

This article is a stub. You can help Wikipedia by fixing it.