Introduction to the Limit of a Function: Learn It 1

  • Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs
  • Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits
  • Understand and use the proper notation for infinite limits and define vertical asymptotes

The Definition of a Limit

We begin our exploration of limits by taking a look at the graphs of the functions

[latex]f(x)=\dfrac{x^2-4}{x-2}, \ \, g(x)=\dfrac{|x-2|}{x-2}[/latex],  and  [latex]h(x)=\dfrac{1}{(x-2)^2}[/latex],

 

which are shown in Figure 1. In particular, let’s focus our attention on the behavior of each graph at and around [latex]x=2[/latex].

"Three 2 and x= -1 for x < 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 / (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two." width="975" height="434"
Figure 1. These graphs show the behavior of three different functions around [latex]x=2[/latex].

Each of the three functions is undefined at [latex]x=2[/latex], but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of [latex]x=2[/latex]. To express the behavior of each graph in the vicinity of [latex]2[/latex] more completely, we need to introduce the concept of a limit.

Intuitive Definition of a Limit

Let’s examine how the function [latex]f(x)=\dfrac{(x^2-4)}{(x-2)}[/latex] behaves as f [latex]x[/latex] approachs [latex]2[/latex]. While [latex]f(x)[/latex] isn’t defined at [latex]x=2[/latex], as [latex]x[/latex] gets closer to [latex]2[/latex] from either side, [latex]f(x)[/latex] approaches [latex]4[/latex].

Mathematically, we say that the limit of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]2[/latex] is [latex]4[/latex]. We express this observation using limit notation as:

[latex]\underset{x \to 2}{\lim}f(x)=4[/latex]

 

This initial exploration into limits leads us to a more formal definition.

Consider the limit of a function at a specific point as the value that the function’s output gets closer to, as the input values approach that point. Assuming such a value exists, we can articulate this concept more precisely with the following definition:

limit definition

For a function [latex]f(x)[/latex] defined over an open interval around a point [latex]a[/latex], possibly excluding [latex]a[/latex] itself, if all function values [latex]f(x)[/latex] get arbitrarily close to some real number [latex]L[/latex] as [latex]x[/latex] approaches [latex]a[/latex], then [latex]L[/latex] is the limit of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex].

[latex]\text{Limit Notation: } \underset{x\to a}{\lim}f(x)=L[/latex]

A more succinct way to understand this definition: As [latex]x[/latex] gets closer to [latex]a[/latex], [latex]f(x)[/latex] gets closer and stays close to [latex]L[/latex]. 


Estimating Limits Using Tables 

We can estimate limits by constructing tables of functional values. 

How to: Evaluate a Limit Using a Table of Functional Values

  1. To find [latex]\underset{x\to a}{\lim}f(x)[/latex], create a table with two sets of [latex]x[/latex]-values: those just less than [latex]a[/latex] and those just more than [latex]a[/latex]. The table below demonstrates what your tables might look like.
    Table of Functional Values for [latex]\underset{x\to a}{\lim}f(x)[/latex]
    [latex]x[/latex] [latex]f(x)[/latex]   [latex]x[/latex] [latex]f(x)[/latex]
    [latex]a-0.1[/latex] [latex]f(a-0.1)[/latex]   [latex]a+0.1[/latex] [latex]f(a+0.1)[/latex]
    [latex]a-0.01[/latex] [latex]f(a-0.01)[/latex] [latex]a+0.01[/latex] [latex]f(a+0.01)[/latex]
    [latex]a-0.001[/latex] [latex]f(a-0.001)[/latex] [latex]a+0.001[/latex] [latex]f(a+0.001)[/latex]
    [latex]a-0.0001[/latex] [latex]f(a-0.0001)[/latex] [latex]a+0.0001[/latex] [latex]f(a+0.0001)[/latex]
    Use additional values as necessary. Use additional values as necessary.
  2. Analyze the [latex]f(x)[/latex] values. If they get closer to a single number as [latex]x[/latex] approaches [latex]a[/latex] from both sides, that’s the limit.
  3. If both sides of [latex]f(x)[/latex] is confirmed. If not, the limit may not exist.
  4. Use the graph of [latex]f(x)[/latex] to verify your results. By plotting the function and zooming in around [latex]x=a[/latex], you can observe if [latex]f(x)[/latex] approaches the limit you calculated. This visual check complements the numerical approach.

Evaluate [latex]\underset{x\to 4}{\lim}\dfrac{\sqrt{x}-2}{x-4}[/latex] using a table of functional values.