Indeterminate form

In calculus, the expressions

$0/0$

$0\cdot \infty$

$1^{\infty }$

$0^{0}$

[others?]

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

$f(x)/g(x)$

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

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

and

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

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

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