1.6: Properties of Limit

Last updated
Save as PDF

Page ID: 5677

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

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

Screen Shot 2020-08-24 at 8.30.38 AM.png

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:

Screen Shot 2020-08-24 at 8.31.42 AM.png

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:

Screen Shot 2020-08-24 at 8.33.13 AM.png

The above results can be encapsulated in the following limit properties:

Basic Limit Properties:

Screen Shot 2020-08-24 at 8.33.44 AM.png

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:

Screen Shot 2020-08-24 at 8.34.25 AM.png

Knowing these properties, allows evaluation of the limits of a wide range of functions.

Take the problem:

Screen Shot 2020-08-24 at 8.37.33 AM.png

Based on the properties above, the limit can be evaluated in the following steps:

Screen Shot 2020-08-24 at 8.38.01 AM.png

Therefore:

Screen Shot 2020-08-24 at 8.38.33 AM.png

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

Screen Shot 2020-08-24 at 8.42.03 AM.png

Based on the properties of limits, the limit can be evaluated in the following steps:

Screen Shot 2020-08-24 at 8.42.56 AM.png

Therefore:

Screen Shot 2020-08-24 at 8.43.31 AM.png

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

Screen Shot 2020-08-24 at 8.44.20 AM.png

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:

Screen Shot 2020-08-24 at 8.44.52 AM.png

Example 2

Evaluate

Screen Shot 2020-08-24 at 9.40.37 AM.png

where f(x) is the rational function

Screen Shot 2020-08-24 at 9.46.41 AM.png

Screen Shot 2020-08-31 at 10.30.41 AM.png

Example 3

Find the following limit if it exists:

Screen Shot 2020-08-31 at 11.07.43 AM.png

Let’s apply the basic quotient rule to evaluate this limit.

Screen Shot 2020-08-31 at 11.19.55 AM.png

Therefore:

Screen Shot 2020-08-31 at 11.32.39 AM.png

Again, the limit result is the same as using direct substitution of x=−4 in the function.

Review

Screen Shot 2020-08-31 at 11.33.07 AM.png

Review (Answers)

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.

Additional Resources

Practice: Properties of Limits