Estimating Limits Graphically and Numerically

Graphically and Numerical Approaches

Graphs and tables are valuable tools for examining the behavior of a function f near, but not at, a given value of x. Finding a limit numerically means using a graphing device (calculator or software) to create a table. Neither of these approaches requires algebraic manipulation or functions that are defined at the point of interest. Both approaches merely suggest a limiting value rather than offering proof of the limit.

image of graph (sin x)/x to estimate limit Examining the graph of $LaTeX: y=\frac{\sin x}{x}$ to estimate $LaTeX: \lim_{x\to 0}=\frac{sin x}{x}$ suggests a limiting value of 1. A numerical approach with values of x very close to 0 also suggests a limiting value of 1. View the presentation below that explores a numerical and graphical approach to this limit.

The next presentation explores additional examples of determining limits graphically.

Limits That Fail to Exist

Often there are situations when limits fail to exist. Three types of behavior are associated with the nonexistence of a limit:

1. The left-hand limit and right-hand limit of the function are not equal.

2. The function increases or decreases without bound as x approaches a given number.

3. The function oscillates between two fixed values as x approaches a given number.

View the examples below of nonexistent limits.

Given the graphs, consider the following:

considerations for graph one limits do not exist in the graph

considerations for graph two text annotation indicator graph: limits do not exist

Several trigonometric functions offer additional examples of limiting values that do not exist. View the presentation below from the beginning until 3:45.

Different Right-Hand and Left-Hand Limits

There are times when the values of a function f tend to different limits as x approaches a number a from the left and right. Click HERE to view the following presentation to explore the behavior of the greatest integer function from graphical and numerical approaches.

Unbounded Behavior

A function f that increases or decreases without bound as x approaches a illustrates unbounded behavior. Use either an online graphing tool or your graphing calculator with an appropriate viewing window to graph these examples to familiarize yourself with functions that increase and/or decrease without bound.

$LaTeX: \lim_{x\to 0}\frac{1}{x^4}$
$LaTeX: \lim_{x\to 2}\frac{1}{x^2-3x+2}$

Oscillating Behavior

When a function alternates between two fixed values, the function is said to exhibit oscillating behavior. View the following presentation to see an example, beginning the video at 3:46.

Use your graphing calculator or an online graphing tool to create a table and examine the behavior of the function $LaTeX: f\left(x\right)=\sin\left(\frac{\pi}{x}\right)$ as x approaches 0. Based on a table of values close to 0, it appears as though the closer we get to 0, the function values approach 0. However, such a conclusion is incorrect. If you view the graph in a standard window, the function values oscillate between -1 and 1. Thus, $LaTeX: \lim_{x\to 0}sin(\frac{\pi}{x})$ does not exist.

Estimating Limits Graphically and Numerically Practice

Estimating Limits Graphically and Numerically: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of how to estimate limits using graphical and numerical methods.

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