# Lesson 11.8: Limits

## Function name and function abbreviation guidelines

The term, limit, is a function name, and lim is the function name abbreviation. Function names and abbreviations are mathematical expressions. The rules that apply to all function names and function abbreviations apply to limits. A space is left between a function name or abbreviation and the expression on which it is acting. Numerals that follow the space require the use of the numeric indicator. The word, limit, or its abbreviation, lim, is often displayed with material directly under it or directly over it. The modification procedure is exactly the same as for sigma notation, that is, the five-step rule for modifications is applied. The steps are the same for either the function name or its abbreviation.

## One sided Function

In functions where the limit approaches from the left or the right the notation will be slightly different. For example, if x were approaching five the notation would be x right arrow five. If x is approaching five from the left side, the previous expression will have a superscripted minus sign such that x right arrow five superscript minus sign. Likewise if x is approaching five from the right side, the expression would have a superscripted plus sign such that x right arrow five superscript plus sign. Hence, if one were trying to find the limit as x approaches five from the right of the function f of x, it would look as follows: limit of subscript x right arrow five superscript plus sign of f of x. Because the x right arrow five superscript plus sign is directly under the limit abbreviation, a baseline indicator, dot five, needs to be placed after the plus or minus sign and before the termination indicator.

In example one, the expression x right arrow seven is displayed in print directly under the abbreviated function name. The arrow is a sign of comparison. The baseline expression, three times x, is the argument of the function; therefore, a space is required between it and the function name.

### Example 1

$\underset{x\to 7}{lim}}3x$

⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠼⠶⠻⠀⠼⠒⠭

### Example 2

$\underset{x\to -2}{lim}}\left({x}^{2}+1\right)$

⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠤⠼⠆⠻⠀⠷⠭⠘⠆⠐⠬⠂⠾

### Example 3

$\underset{n\to \infty}{lim}}{a}_{n$

⠐⠇⠊⠍⠩⠝⠀⠫⠕⠀⠠⠿⠻⠀⠁⠰⠝

### Example 4

$\underset{n\to 0}{lim}}f\left(x\right)$

⠐⠇⠊⠍⠩⠝⠀⠫⠕⠀⠼⠴⠻⠀⠋⠷⠭⠾

### Example 5

$\frac{{\displaystyle \underset{x\to 1}{lim}}f\left(x\right)}{{\displaystyle \underset{x\to 1}{lim}}g\left(x\right)}$

⠹⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠼⠂⠻⠀⠋⠷⠭⠾⠌⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠼⠂⠻⠀⠛⠷⠭⠾⠼

### Example 6

$\underset{x\to {4}^{-}}{lim}f\left(x\right)$

⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠼⠲⠘⠤⠐⠻⠀⠋⠷⠭⠾

### Example 7

$\underset{x\to {4}^{+}}{lim}f\left(x\right)$

⠐⠇⠊⠍⠩⠭⠀⠫⠕⠀⠼⠲⠘⠬⠐⠻⠀⠋⠷⠭⠾