How to write radical, limit and quadrature symbols in LaTeX

Key points of this article

Use \sqrt{} for the radical, \sqrt[n]{} for the nth root, and the curly braces should surround the entire divination.
The limit is written as \lim_{x\to 0}. Do not type the letter lim directly, otherwise the spacing and font will be incorrect.
Use \int for quadrature, upper and lower limits as \int_{a}^{b}, and use \iint or \iiint for multiple quadratures.
Infinity is \infty, the trend is written as \to, and the upper and lower limit positions of inline formulas and independent formulas are different.

When writing papers, doing homework, or organizing lecture notes, many people are stuck in the same place: how to enter the root sign, limit and quadrature symbols in the formula, and where should the upper and lower limits be placed. These symbols look simple, but once the position is wrong, the formula will either compile with an error or the formatting will be ugly, which can be seen by reviewers or teachers at a glance.

This article is written for readers who can already write basic LaTeX formulas but are unfamiliar with operator symbols. It discusses the root sign, lim, quadrature symbol and infinity together, focusing on how to write commands, the difference between the positions of upper and lower limits within a line and in independent formulas, and how to check after writing. After reading, you should be able to independently write common limit, quadrature, and radical expressions.

What commands are used for root, limit, product and infinity? One sentence conclusion

First put these types of symbols into the mathematical environment, and then input them with the corresponding commands; the upper and lower limits are controlled by the environment and \limits and \nolimits.

your situation	Recommended practices	reason
Ordinary square root, such as x under the root sign	\sqrt{x}	Curly braces enclose the content being squared. It is recommended to keep it even if there is only one character.
Cube root or nth root	\sqrt[3]{x}, \sqrt[n]{x}	Write the number in square brackets. The number can be letters.
Find the limit, tend to a certain value	\lim_{x\to 0}	\lim is the standard operator, \to is the arrow, and the subscript is written after _
Find the product, there are upper and lower limits	\int_{a}^{b}	Use _ for the lower limit and ^ for the upper limit, both enclosed in curly braces.
Great size	\infty	Do not write inf or ∞ as hand-typed characters. \infty is more standard for typesetting.

This table covers most common situations. There is no position problem for radicals and infinity, as long as you pay attention to the curly braces and the mathematical environment; the upper and lower limit positions of limits and quadrature are affected by the display mode, which will be expanded in the next section.

Before you start: Confirm these points first

Before you start, determine the environment where the formula is located, the macro package used, and the output target, which can save you a lot of detours.

Whether the formula is displayed inline or independently. Inline formulas are written in the middle of a paragraph of text, and independent formulas occupy a separate line. For the same \lim, the upper and lower limits will be pressed into small characters in the lower right corner when inline, and will be placed directly below when independent. The judgment method is simple: see if it is embedded in the sentence.
Confirm formula in math environment. The square root, lim, and quadrature symbols are only available in math mode. Inline formulas are wrapped with $, stand-alone formulas are wrapped with \[ and \], or use the equation environment. Forgot packages are a common source of errors.
Remember the difference between \lim and lim. If you type the three letters l, i, and m directly, the typesetting will be treated as the multiplication of three italicized variables, and the spacing will be incorrect. \lim is the standard limit operator, and the font and spacing match it.
Prepare common basic macro packages. \sqrt, \lim, \int, and \infty are all built-in commands of LaTeX and do not require additional macro packages. Only if you need more complex formatting, such as \dfrac or a custom quadrature symbol, do you need to check the package settings in the document leader area.
Determine the final usage scenario. If you want to translate a LaTeX source code package with formulas, the formula itself should remain the same, focusing on the explanatory text in the translation. If you are writing formulas in a web editor, you can directly enter the LaTeX command and view the rendering results.

Write out the steps of root sign, limit and quadrature one by one in LaTeX

Write in the order of "first wrap the environment, then write the symbols, and finally add the lower limit". Compile or preview each symbol written.

Create a new mathematical environment. Write the content between $ and $ within a line, and write the content between \[ and \] for independent formulas.
Write the root sign. Ordinary square root input \sqrt{radicand mode}, for example \sqrt{x^2+1}. When the open method is longer, the entire expression is enclosed in curly braces.
Write the nth root. Add square brackets after \sqrt to write the number, for example \sqrt[3]{x}. When the frequency is a letter or multiple digits, also put it in square brackets.
Write limits. Type \lim, followed by _ to write the subscript, and then wrap the complete trend condition in curly braces, for example \lim_{x\to 0}. Use \to for arrows, not ->.
Write infinity. When going to infinity, write the subscript as \lim_{n\to\infty}. \infty can be used directly in any mathematical environment.
Write the product. Enter \int, using _{a} for the lower limit and ^{b} for the upper limit, for example \int_{a}^{b} f(x)\,dx. Add a \, thin space in front of the differential dx to make it visually clearer.
Handle multiple quadratures. Use \iint for double quadrature and \iiint for triple quadrature. The upper and lower limits are written in the same way as single quadrature.
Finally, check whether the curly braces are paired and compile them once. Make sure the upper and lower limits are not crowded together or in the wrong position.

Where are the upper and lower limits placed? The difference between inline and stand-alone formulas

\lim and \sum default to placing the upper and lower limits directly above and directly below in independent formulas, and in small characters on the right side of the line; \int defaults to placing both upper and lower limits on the right side.

This is where many people get confused. The following comparison can help you judge when to use the default writing method and when to manually modify it.

symbol	Default position of inline formulas	Default position for stand-alone formulas	When you want to force a change
\lim	Small print in the lower right corner	directly below	If you want to place it directly below the line, use \lim\limits_{x\to 0}
\int	right side	right side	If you want to align the top and bottom, use \int\limits_{a}^{b}
\sum	Small print on the right	directly above and below	If you want to place it at the top and bottom of the line, use \sum\limits_{i=1}^{n}

If you want the limits and products in the inline formula to look consistent with the independent formula, you can add \limits to force them to be placed up and down; if the upper and lower limits in the independent formula occupy too many lines, you can use \nolimits to move them to the right. Both writing methods do not change the mathematical meaning, only adjust the layout.

Check each item after completion

After compilation or preview, check one by one whether the symbols are displayed correctly, whether the upper and lower limits are in reasonable positions, and whether the curly braces are balanced.

Check items	How to see	What to do if there is a problem
Does compilation report an error?	Check whether Missing } inserted or Undefined control sequence appears in the log.	Most of the errors are caused by unpaired curly braces or misspelled commands. Check and modify them one by one.
Is the root number range correct?	Does the root horizontal line cover the entire quilt pattern?	Put the open method completely in curly braces, such as \sqrt{x+1} instead of \sqrt x+1
Extreme and lim fonts	lim should appear as the upright standard operator	If it is displayed in italics l i m, it means that the backslash is missing and changed to \lim
Upper and lower limit positions	Is the limit directly below, and is it customary to find the upper and lower limits of product?	Add \limits or \nolimits according to the table in the previous section
Upper and lower limit length	Do subscripts and superscripts squeeze each other?	Multi-character subscripts must be enclosed in curly braces, such as _{n\to\infty}
infinity and arrows	Are \infty and \to displayed normally?	Check that you are in a math environment and confirm the command spelling

If the formula content comes from translation or conversion, the mathematical meaning must also be checked against the original text. Machine translation or format conversion may change symbols or upper and lower limits. It is recommended to compare important formulas with the original manuscript one by one.

📊 Core comparison

Inline formulas are suitable for short expressions, and independent formulas are suitable for limits and quadratures that need to be highlighted. The symbols of the two are written in the same way, but the layout is different.

Writing method	Example	Upper and lower limit positions	Applicable scenarios
Inline formula default	$\lim_{x\to 0}\frac{\sin x}{x}=1$	Small print in the lower right corner	Short expressions, don’t want to mess with paragraph spacing
Independent formula default	\[ \lim_{x\to 0}\frac{\sin x}{x}=1 \]	directly below	Limits or quadratures that need to be highlighted
Forced uniformity within the industry	$\lim\limits_{x\to 0}\frac{\sin x}{x}=1$	directly below	Inline formulas must be consistent with the independent formula layout

❓ Frequently Asked Questions (FAQ)

1. Can the curly braces of the root sign be omitted?

Single characters such as \sqrt When the curly braces are omitted, the root sign only acts on the immediately following character, and the result is often not as expected.

2. Why did I write \lim but the subscript was not placed directly below?

This is usually because the formula is in inline mode. When inline, the upper and lower limits of \lim are placed in the lower right corner by default. If you need directly below, add \limits after \lim.

3. Why are the upper and lower limits of product always on the right?

\int uses \nolimits by default, with upper and lower limits placed on the right. This is the normal layout of quadrature symbols. If you need to put it above and below, write \int\limits_{a}^{b}, but this is usually only done in stand-alone formulas.

4. What are the common mistakes in writing infinity symbols and arrows?

For infinity, write \infty, not inf; for arrows, write \to, not -> or →. Unicode symbols pasted directly may not display properly in some compilation environments.

5. Can the LaTeX source code package with formulas be translated?

Can. OmniDoc supports document translation of LaTeX source code packages. The translation retains formulas and typesetting. The translated explanatory text can be bilingual or only the translation can be output. After translation, you still need to check whether the formula is consistent with the original text.

📝 Summary

Use \sqrt for the radical, \lim for the limit, \int for the product, and \infty for the infinity. These four commands cover most common formulas. When writing, pay attention to the inline and independent differences between curly braces, mathematical environments, and upper and lower limits, and check them one by one after compilation to avoid most typesetting problems.

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