# 1.6: Properties of Limit

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

You're familiar with the idea of a limit of a function, and that some limits are computed using numerical and graphical methods. Limits can also be evaluated using the properties of limits. How would you find without using a graph or using a table of values?

## Properties of Limits

Let’s begin with some observations about limits of some simple functions. Consider the following limit problems: We note that each of these functions is defined for all real numbers. If we apply our intuition for finding the limits we conclude correctly that: The above results can be encapsulated in the following limit properties:

Basic Limit Properties: Many functions can be expressed as the sums, differences, products, quotients, powers and roots of other more simple functions. The following properties are also useful in evaluating limits:

More Basic Limit Properties: Knowing these properties, allows evaluation of the limits of a wide range of functions.

Take the problem: Based on the properties above, the limit can be evaluated in the following steps: Therefore: Note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of x=2 in the function.

Now, evaluate Based on the properties of limits, the limit can be evaluated in the following steps: Therefore: Again, note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of x=2 in the function.

## Examples

### Example 1

Earlier, you were asked to find without using a graph or a table of values. This is a case where direct substitution to evaluate the limit gives the indeterminate form 0/0. But, reducing the fraction and then applying the basic limit properties above, we can evaluate the limit: ### Example 2

Evaluate where f(x) is the rational function  ### Example 3

Find the following limit if it exists: Let’s apply the basic quotient rule to evaluate this limit. Therefore: Again, the limit result is the same as using direct substitution of x=−4 in the function.

## Review To see the Review answers, open this PDF file and look for section 2.3.

## Vocabulary

Term Definition
indeterminate In mathematics, an expression is indeterminate if it is not precisely defined. There are seven indeterminate forms: 0/0,0⋅∞,∞∞,∞−∞,00,∞0, and 1.
limit A limit is the value that the output of a function approaches as the input of the function approaches a given value.