Limits with Absolute Value

Limits with Absolute Value
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  • 0:38 How Limits Work
  • 1:33 Functions Without A Limit
  • 2:40 One-Sided Limits
  • 3:12 Piecewise Functions
  • 4:07 Examples
  • 4:56 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

While limits in calculus can be evaluated in many different ways, those involving absolute value can be especially tricky. In this lesson, we'll use plenty of examples to show you how to compute limits in calculus with absolute value using a graph or algebra.

Limit in Calculus

A limit in calculus is the likely height of a function, or the relationship between input and output, at a known value of x. We express the limit of a function as f(x) and define it as the projected y value of f(x) as the x values get closer and closer to any real number c. Keep in mind that the function doesn't actually have to reach the projected height or limit. When evaluating complex limits in calculus, we sometimes use absolute value, or how far away a number on a number line is located from zero, regardless of its direction.

How Limits Work

Let's start with an easy problem, which will give you a chance to see how limits work in general. Consider the function f(x) = |x + 1| - 2, as shown in this graph.

y = |x + 1| - 2
Graph of f(x)=|x+1|-2

What is the limit of f as x approaches 0?

Notice that when x is near the value 0, the graph seems to have y values that are nearer to -1. Based on the graph, we can assume the limit is equal to -1. To find the precise limiting value, we plug x = 0 into the function, as shown here.

Limit example worked out

In this function, the limit of f as x approaches any real number, c, can be found by plugging in x = c because f is continuous with no breaks or no holes. So, can we always use plug in to get the limit? If that were true, this would be a very short lesson indeed! Let's explore what issues might come up when calculating limits with absolute value.

Functions Without a Limit

This graph shows the function f(x) = |x|/x. What is the limit as x approaches 0?

y = |x| / x
Graph of y = |x|/x

This time, plugging in x = 0 results in nonsense. |0|/0 = 0/0 is what we call an indeterminate form. As the rules for calculating limits do not allow us to divide by zero, let's go back to the graph for more insight.

The graph behaves differently on the left-hand and right-hand sides of x = 0. Starting from the left, the graph maintains a steady height of y = -1. As x passes through 0, it suddenly jumps to a height of y = 1, a phenomenon known as jump discontinuity.

According to the graph, the limit as x approaches 0 fails to exist. As x gets closer to 0, we don't see the function heading towards a y value. In this case, we write it like this:

Limit of |x|/x does not exist at 0

The letters DNE stand for does not exist. This is a fairly standard way to indicate a non-existent limit.

One-Sided Limits

Some limit problems only ask for the limiting value as x approaches from the left-hand or right-hand side. For example, if we only asked about the limit of |x| / x as x approaches 0 from the left, then the answer would be -1. Similarly, the limit of |x| / x as x approaches 0 from the right is equal to 1. While we can see the values in this graph, what happens if you don't have a graph? Fortunately, we can use algebra to work out these limits.

Piecewise Function

Sometimes, absolute value functions, f(x) = |x|, look like two different lines stuck together at a sharp corner. We can approach an absolute value function as a piecewise function, in which a piece of y = -x is joined to a piece of y = x.

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