Limit

limit

$\lim_{ x \to +0 } \frac{1}{x} = \infty$

\lim_{ x \to +0 } \frac{1}{x} = \infty

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It is derived from the concept of a limit.

limit large

$\displaystyle \lim_{ n \to \infty } f_n(x) = f(x)$

\displaystyle \lim_{ n \to \infty } f_n(x) = f(x)

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Adding "\displaystyle" causes the expression to be displayed in a larger format. Subscripts will be positioned below the "lim" symbol.

limit superior

$\limsup_{ n \to \infty } a_n$

\limsup_{ n \to \infty } a_n

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It is derived from the limit superior.

limit superior simple

$\varlimsup_{ n \to \infty } a_n$

\varlimsup_{ n \to \infty } a_n

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limit inferior

$\liminf_{ n \to \infty } a_n$

\liminf_{ n \to \infty } a_n

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It is derived from the limit inferior.

limit inferior simple

$\varliminf_{ n \to \infty } a_n$

\varliminf_{ n \to \infty } a_n

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limit superior sample

$\begin{eqnarray} \varlimsup_{ n \to \infty } a_n = \lim_{ n \to \infty } \sup_{ k \geqq n } a_k \end{eqnarray}$

\begin{eqnarray}
\varlimsup_{ n \to \infty } a_n
 = \lim_{ n \to \infty } \sup_{ k \geqq n } a_k
\end{eqnarray}

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This is an example of the limit superior of a sequence.

limit inferior sample

$\begin{eqnarray} \varliminf_{ n \to \infty } A_n = \bigcup_{ n = 1 }^{ \infty } \bigcap_{ k = n }^{ \infty } A_k = \bigcup_{ n \in \mathbb{ N } } \bigcap_{ k \geqq n } A_k \end{eqnarray}$

\begin{eqnarray}
\varliminf_{ n \to \infty } A_n
 = \bigcup_{ n = 1 }^{ \infty } \bigcap_{ k = n }^{ \infty } A_k
 = \bigcup_{ n \in \mathbb{ N } } \bigcap_{ k \geqq n } A_k
\end{eqnarray}

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Here is an example of the limit inferior of a set.

Big O notation

$\mathcal{O}$

\mathcal{O}

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In some cases, the symbol for Landau notation is represented using the letter "O" in calligraphy fonts.