Limit (lim) in LaTeX
\lim typesets the upright lim operator; a subscript written with \to states what the variable approaches, as in \lim_{x \to a}. For a limit at infinity write \lim_{x \to \infty}.
LaTeX command
\lim_{x \to a}Not the symbol you meant? Draw it to find the command →
\lim is a named operator that sets lim upright with correct spacing. The point the variable tends to goes in a subscript, using \to (or \rightarrow) for the arrow: \lim_{x \to a} f(x). For a limit at infinity write \lim_{x \to \infty}, and for a one-sided limit add a sign to the target, or .
Placement of the subscript depends on mode. In displayed math the x to a sits centred beneath lim, which is the standard book form. Inline it appears as a subscript to the right to save vertical space. You can force the underneath position inline by wrapping the expression in \displaystyle.
Related operators follow the same pattern: \limsup and \liminf for the upper and lower limits, \sup and \inf, and \max and \min. All are predefined so they set upright with proper spacing. For a name of your own use \operatorname, or \operatorname* to put the limits underneath.
Other forms
\limsuplimit superior\liminflimit inferiorExamples
$\lim_{x \to 0} \frac{\sin x}{x} = 1$a standard limit
$\lim_{x \to \infty} \frac{1}{x} = 0$a limit at infinity
$\lim_{x \to 0^{+}} \ln x = -\infty$a one-sided limit
$\lim_{n \to \infty} \left(1 + \tfrac{1}{n}\right)^n = e$the definition of e
$\limsup_{n \to \infty} a_n$the limit superior
Tips and common mistakes
- Always use
\lim, not the plain letters lim, so the operator sets upright with correct spacing. - Put the target in a subscript with
\to:\lim_{x \to a}; use\inftyfor limits at infinity. - For one-sided limits mark the target: (from above) or (from below).
- In displayed math the subscript centres below lim; inline it moves to the side. Wrap in
\displaystyleto force the under-set form inline. \limsupand\liminfare their own commands; do not build them from\limplus text.
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Convert your maths to LaTeX →FAQ
How do I write a limit in LaTeX?
Use \lim with the target in a subscript, for example . \to draws the arrow.
How do I write a limit as x approaches infinity?
Put \infty in the subscript: .
Why is the x to a beside lim instead of underneath?
That is the inline form, which saves vertical space. In displayed math it centres beneath lim; you can force that inline by wrapping the expression in \displaystyle.