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Limit in LaTeX

Write \lim followed by the condition as a subscript: \lim_{x \to a} f(x). \to draws the arrow. In a displayed equation the condition sits under “lim”; inside a line of text it moves to the side unless you add \limits.

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Limit commands

  • \lim_{x \to a} f(x)
    lim x → a f ( x )

    Limit as x approaches a.

  • \lim_{x \to \infty} f(x)
    lim x → ∞ f ( x )

    Limit at infinity.

  • \lim_{x \to a^{+}} f(x)
    lim x → a + f ( x )

    Right-hand limit.

  • \lim_{x \to a^{-}} f(x)
    lim x → a − f ( x )

    Left-hand limit.

  • \lim_{n \to \infty} a_n
    lim n → ∞ a n

    Limit of a sequence.

  • \limsup_{n \to \infty} a_n
    lim sup n → ∞ a n

    Limit superior (amsmath).

  • \liminf_{n \to \infty} a_n
    lim inf n → ∞ a n

    Limit inferior (amsmath).

  • \lim\limits_{x \to 0}
    lim x → 0

    Keeps the condition under lim inside running text.

  • x \searrow 0
    x ↘ 0

    Decreasing approach, used by some authors instead of 0⁺.

Try it live

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Templates

Limit examples

Each result below is drawn from the LaTeX under it. Copy the code, edit it on this page, or open it in the full editor.

  • Classic limit

    lim x → 0 sin ⁡ x x = 1
    LaTeX
    \lim_{x \to 0} \frac{\sin x}{x} = 1
    Edit hereOpen in editor

  • Definition of e

    lim n → ∞ ( 1 + 1 n ) n = e
    LaTeX
    \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e
    Edit hereOpen in editor

  • One-sided limits

    lim x → 0 + 1 x = ∞ , lim x → 0 − 1 x = − ∞
    LaTeX
    \lim_{x \to 0^{+}} \frac{1}{x} = \infty, \quad \lim_{x \to 0^{-}} \frac{1}{x} = -\infty
    Edit hereOpen in editor

  • Derivative as a limit

    f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h
    LaTeX
    f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
    Edit hereOpen in editor

  • Limit superior

    lim sup n → ∞ sin ⁡ n = 1
    LaTeX
    \limsup_{n \to \infty} \sin n = 1
    Edit hereOpen in editor

  • Two-variable limit

    lim ( x , y ) → ( 0 , 0 ) x y x 2 + y 2 = 0
    LaTeX
    \lim_{(x, y) \to (0, 0)} \frac{x y}{\sqrt{x^2 + y^2}} = 0
    Edit hereOpen in editor

  • Limit in running text

    lim x → ∞ 1 x = 0
    LaTeX
    \textstyle \lim\limits_{x \to \infty} \frac{1}{x} = 0

    With \limits the condition stays under “lim” even at text size.

    Edit hereOpen in editor

How to write a limit step by step

Use \lim, not the letters lim. The command sets “lim” upright, spaces it as an operator and knows where the subscript belongs; typed letters come out as italic variables l·i·m.

Put the whole condition in braces after _: \lim_{x \to 0}. \to and \rightarrow draw the same arrow; \to is shorter and the usual choice. For one-sided limits add ^{+} or ^{-} to the value: x \to 0^{+}.

Displayed equations put the condition under “lim”. Inline, LaTeX moves it to the right to keep the line height; \lim\limits_{…} overrides that for one limit.

LaTeX
\[
  \lim_{x \to 0} \frac{\sin x}{x} = 1
\]
In text: $\lim\limits_{n \to \infty} \frac{1}{n} = 0$.

Your own limit-like operators

For names amsmath does not define, such as ess sup or arg max, declare an operator with limits: \DeclareMathOperator*{\esssup}{ess\,sup} in the preamble, or \operatorname*{arg\,max}_{x} in place. The star is what puts the subscript underneath in displays.

Common mistakes

  • Without the backslash “lim” is three italic variables and the subscript attaches only to the m.

    Instead of
    lim_{x \to 0} f(x)
    Write
    \lim_{x \to 0} f(x)

  • Only x is in the subscript; the arrow and 0 end up on the main line.

    Instead of
    \lim_x \to 0 f(x)
    Write
    \lim_{x \to 0} f(x)

  • -> is set as a minus sign and a greater-than sign, not an arrow.

    Instead of
    \lim_{x -> 0}
    Write
    \lim_{x \to 0}

In Overleaf, MathJax and Word

  • Overleaf and pdfLaTeX

    \lim works in any document. \limsup, \liminf, \operatorname* and \DeclareMathOperator* need \usepackage{amsmath}.

  • MathJax and KaTeX

    MathJax and KaTeX render \lim, \limsup, \liminf, \limits and \operatorname* without setup. Inline math in Markdown ($…$) puts the condition to the side, just as LaTeX does.

  • Microsoft Word

    Word has a “lim” template under Function, but pasted LaTeX is not converted reliably. Write the limit here and use Copy for Word to paste an editable Word equation.

Limit in LaTeX: questions

How do I put x → 0 under lim?
Write it as a subscript: \lim_{x \to 0}. It appears underneath in displayed equations; for inline math add \limits: \lim\limits_{x \to 0}.
How do I write a limit approaching infinity?
Use \infty: \lim_{x \to \infty} f(x). For negative infinity write x \to -\infty.
How do I write a one-sided limit?
Add a plus or minus superscript to the value: \lim_{x \to a^{+}} for the right-hand limit and \lim_{x \to a^{-}} for the left-hand one.
Is \to the same as \rightarrow?
Yes, both draw →. \to is the conventional choice in limits because it is short and says what it means.

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