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.

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

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