# Lesson 10.1: Limits

## Review

Grade 1 mode is turned on by grade 1 indicators and by the numeric indicator. When grade 1 mode is turned on by the numeric indicator, it is turned off by a space, hyphen, or dash. When complex algebraic equations contain indicators such as superscript, subscript, fractions, radicals, and/or letters standing alone, it is best to enclose them in grade 1 passage indicators to ensure the symbols are well defined without the need for grade 1 symbol indicators. When a complex expression is comprised of a single symbol sequence, a grade 1 word indicator will be enough to ensure that the various indicators are well defined without the need for grade 1 symbol indicators.

Braille has four level change indicators: level change up (superscript), level change down (subscript), expression directly above, and expression directly below. The scope of a level change indicator is the next item. Braille grouping indicators must be used to enclose an expression that includes more than one item. A sign of comparison that is not on the base line should be unspaced from adjacent symbols.

## Explanation

The term limit is used in mathematics to mean approaching and is indicated by the abbreviation lim or lm. It is the value that a function or sequence approaches as the input or index approaches some value. In a limit expression, material is usually displayed directly below the word limit or its abbreviation, with a value to its right.

Limit is brailled as it appears in print, followed by the expression below and then the element(s) to the right of the limit. Use the expression directly below indicator for material displayed directly below the limit. The extent of the expression directly below indicator is the next item. Braille grouping indicators are required if the expression extends beyond one item. The element directly to the right of the limit is brailled as it is shown in print.

Arrows are signs of comparison and should usually be spaced. An exception is when they are written below the limit function.

### Example 1

$\underset{x\to 1}{lim}}\mathrm{fx$

⠰⠰⠇⠊⠍⠨⠢⠣⠭⠳⠕⠼⠁⠜⠋⠭

### Example 2

$\underset{x\to -1.5}{lim}}f\left(x\right)$

⠰⠰⠇⠊⠍⠨⠢⠣⠭⠳⠕⠐⠤⠼⠁⠲⠑⠜⠋⠐⠣⠭⠐⠜

### Example 3

$\underset{n\to \mathrm{\infty}}{lim}}\left(\frac{1}{n}\right)=0$

⠰⠰⠇⠊⠍⠨⠢⠣⠝⠳⠕⠼⠿⠜⠐⠣⠷⠼⠁⠨⠌⠝⠾⠐⠜⠀⠐⠶⠀⠼⠚

### Example 4

$\underset{a\to 1}{lim}}f\left(x\right)$

⠰⠰⠇⠊⠍⠨⠢⠣⠁⠳⠕⠼⠁⠜⠋⠐⠣⠭⠐⠜

### Example 5

$\underset{x\to {2}^{-}}{lim}f\left(x\right)$

⠰⠰⠇⠊⠍⠨⠢⠣⠭⠳⠕⠼⠃⠔⠐⠤⠜⠋⠐⠣⠭⠐⠜

### Example 6

$\underset{x\to {2}^{+}}{lim}f\left(x\right)$

⠰⠰⠇⠊⠍⠨⠢⠣⠭⠳⠕⠼⠃⠔⠐⠖⠜⠋⠐⠣⠭⠐⠜

### Example 7

$\underset{n\to \mathrm{\infty}}{lim}}{a}_{n}=L$

⠰⠰⠰⠇⠊⠍⠨⠢⠣⠝⠳⠕⠼⠿⠜⠁⠢⠝⠀⠐⠶⠀⠠⠇⠰⠄