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The Graph Method

Author: Sophia
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what's covered
In this lesson, you will evaluate limits by using the graph of a function. Specifically, this lesson will cover:

Table of Contents

1. Defining Limit Notation

Consider the function f open parentheses x close parentheses equals fraction numerator x squared minus 1 over denominator x minus 1 end fraction.

Notice that f open parentheses x close parentheses is undefined when x equals 1. However, we still may want to analyze the behavior of f open parentheses x close parentheses around x equals 1. The mathematical tool used to do this sort of analysis is called a limit.

big idea
limit as x rightwards arrow a of f open parentheses x close parentheses equals L means “the limit of f open parentheses x close parentheses as x gets closer to a is equal to L”. In other words, as x gets closer to a, the value of f open parentheses x close parentheses gets closer to L. We call L the limit of the function f open parentheses x close parentheses.

To see how this works graphically, shown below is the graph of f open parentheses x close parentheses equals fraction numerator x squared minus 1 over denominator x minus 1 end fraction.

A graph with the x-axis and y-axis ranging from −6 to 6. A line slants upward from the third quadrant and passes through the x-axis at (−1, 0) and up through the first quadrant, but also has an open circle at (1, 2).

Notice that there is a hole in the graph at the point open parentheses 1 comma space 2 close parentheses comma indicating that the graph of f open parentheses x close parentheses is a line, but excludes the point open parentheses 1 comma space 2 close parentheses.

Since f open parentheses x close parentheses is undefined when x equals 1 comma we analyze the behavior of f open parentheses x close parentheses by using limits.

That is, we want to evaluate limit as x rightwards arrow 1 of f open parentheses x close parentheses or more specifically, limit as x rightwards arrow 1 of fraction numerator x squared minus 1 over denominator x minus 1 end fraction.

By examining the graph, it appears that as x gets closer and closer to 1, f open parentheses x close parentheses gets closer and closer to 2. Thus, we can write limit as x rightwards arrow 1 of fraction numerator x squared minus 1 over denominator x minus 1 end fraction equals 2.

term to know
Limit
The value that a function f open parentheses x close parentheses approaches as x gets closer to a specified number.

2. Using Graphs to Evaluate Limits

We can use the information from a graph to evaluate a limit.

EXAMPLE

Consider the graph of some function y equals f open parentheses x close parentheses.
A graph with an x-axis ranging from 0 to 4 and a y-axis ranging from −1 to 2. The graph represents the function y equals f(x) and has a curve and a line. The curve rises from the left of the y-axis, passes through the point (0, 1), peaks slightly, then passes through an open circle at (1, 1), dips slightly downward, and ends at another open circle at (2, 1). The line segment begins from a closed dot at (2, 2), extends to an open circle at (3, 2), bends sharply to a closed dot at (4, 0), and then slants upward from this point. A closed dot is marked at (3, 1).
We can say the following:

Statement Description
limit as x rightwards arrow 0 of f open parentheses x close parentheses equals 1 As x gets closer to 0, f open parentheses x close parentheses gets closer to 1.
limit as x rightwards arrow 1 of f open parentheses x close parentheses equals 1 As x gets closer to 1, f open parentheses x close parentheses gets closer to 1.
limit as x rightwards arrow 2 of f open parentheses x close parentheses does not exist. As x gets closer to 2 from the left (values smaller than 2), f open parentheses x close parentheses gets closer to 1. However, as x gets closer to 2 from the right (values larger than 2), f open parentheses x close parentheses gets closer to 2.

Since f open parentheses x close parentheses approaches two different values, as x approaches 2, we say the limit does not exist.
limit as x rightwards arrow 3 of f open parentheses x close parentheses equals 2 As x gets closer to 3, f open parentheses x close parentheses gets closer to 2. Note that the actual value of f open parentheses 3 close parentheses is 1 (closed dot at x equals 3), but the limit tells us what is happening as we get closer and closer to 3, not what is happening right at 3.
limit as x rightwards arrow 4 of f open parentheses x close parentheses equals 0 As x gets closer to 4, f open parentheses x close parentheses gets closer to 0.

try it
Consider the graph pictured below.

A graph with an x-axis ranging from 0 to 4 and a y-axis ranging from −1 to 2. The graph represents the function y equals f(x) and has two curves. One curve rises from the left of the y-axis, passes through the point (0, 1.75), peaks slightly just before x equals 1, then passes through a closed dot at (1, 2), dips slightly downward, rises again to connect to an open circle at (2, 2), rises slightly to a peak around the point (2.5, 2.25), then falls slightly and ends at a closed dot at (3, 2). The other curve begins from an open circle at (3, 1), rises to a peak at an open circle at (4, 2), and then declines. A closed dot is marked at (2, 1).

Evaluate the function as x approaches 1.
limit as x rightwards arrow 1 of f open parentheses x close parentheses equals 2
Evaluate the function as x approaches 2.
limit as x rightwards arrow 2 of f open parentheses x close parentheses equals 2
Evaluate the function as x approaches 3.
limit as x rightwards arrow 3 of f open parentheses x close parentheses does not exist
Evaluate the function as x approaches 4.
limit as x rightwards arrow 4 of f open parentheses x close parentheses equals 2

summary
In this lesson, you learned about defining limit notation, or how the limit of a function is used to determine the behavior (or value) a function f open parentheses x close parentheses approaches as x gets closer to some value. You also learned that you can use the information from a graph to evaluate a limit.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 1 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Terms to Know
Limit

The value that a function f open parentheses x close parentheses approaches as x gets closer to a specified number.

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