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

f(x)=x24x2, g(x)=|x2|x2,  and  h(x)=1(x2)2,

 

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

"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 x=2.

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

Intuitive Definition of a Limit

Let’s examine how the function f(x)=(x24)(x2) behaves as f x approachs 2. While f(x) isn’t defined at x=2, as x gets closer to 2 from either side, f(x) approaches 4.

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

limx2f(x)=4

 

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

Limit Notation: limxaf(x)=L

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


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 limxaf(x), create a table with two sets of x-values: those just less than a and those just more than a. The table below demonstrates what your tables might look like.
    Table of Functional Values for limxaf(x)
    x f(x)   x f(x)
    a0.1 f(a0.1)   a+0.1 f(a+0.1)
    a0.01 f(a0.01) a+0.01 f(a+0.01)
    a0.001 f(a0.001) a+0.001 f(a+0.001)
    a0.0001 f(a0.0001) a+0.0001 f(a+0.0001)
    Use additional values as necessary. Use additional values as necessary.
  2. Analyze the f(x) values. If they get closer to a single number as x approaches a from both sides, that’s the limit.
  3. If both sides of f(x) is confirmed. If not, the limit may not exist.
  4. Use the graph of f(x) to verify your results. By plotting the function and zooming in around x=a, you can observe if f(x) approaches the limit you calculated. This visual check complements the numerical approach.

Evaluate limx4x2x4 using a table of functional values.