Indeterminate form

In calculus, the expressions

${\displaystyle {0 \over 0}}$

${\displaystyle {\infty \over \infty }}$

${\displaystyle 0\cdot \infty }$

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

${\displaystyle 0^{0}}$

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

${\displaystyle \infty -\infty }$

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

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

can approach any real number or ∞ or −∞, or fail to converge to any point on the extended real number 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

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

Direct substitution of the number that x approaches into either of these functions leads to the indeterminate form 0/0, but both limits actually exist and are 1 and 14 respectively.

The indeterminate form does not imply the limit does not exist. Algebraic elimination or L'Hôpital's rule can be used to simplify the expression so the limit can be more easily evaluated.