Limit in LaTeX
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A limit is written with \lim, and the condition goes in the subscript with the arrow \to. In display math the condition is placed under the word lim.
Limit of a function
\lim_{x \to 0} \frac{\sin x}{x} = 1\lim is an operator: upright, with the condition placed underneath in display math. \to draws the arrow read as "tends to".
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Limit of a sequence
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e\infty is the infinity sign. The \left( \right) pair grows with the fraction inside, which plain brackets would not do.
One-sided limit
\lim_{x \to 0^{+}} \ln x = -\inftyThe sign in the superscript, 0^{+} or 0^{-}, says from which side x approaches. Some textbooks write x \searrow 0 for a limit from above.
Limit inside a sentence
$\lim_{x \to 0} f(x)$In running text the condition moves to the side of lim to keep the line height even. \lim\limits_{x \to 0} forces it underneath when you really need it.
Several conditions
\lim_{\substack{x \to 0 \\ y \to 0}} f(x, y)\substack stacks several conditions under lim, as in the limit of a function of two variables.
Upper and lower limits
\limsup_{n \to \infty} a_n, \quad \liminf_{n \to \infty} a_nThe limit superior and inferior have their own operators, \limsup and \liminf, set upright with the condition underneath just like \lim.
Common mistakes
- Writing lim without a backslash: it is set in italics like the product l·i·m.
- Typing -> instead of \to: in math mode you get a minus sign followed by a greater-than sign.
- The condition on the line, as in \lim x \to 0: it loses the subscript and reads like a product.
See also
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