Lesson 1.3: Utilizing the ε-δ Definition and Limits that Do Not Exist

Example 1.1


Use the ε-δ definition to show

(Feel free to try this one yourself before looking at the answer to see if you can get it.)

Solution: What we are asking here is to show that for each ε > 0, there exists a δ > 0 such that 0 < |x - 2| < δ implies that |x² - 4| < ε. The first thing to notice is that |x² - 4| = |x - 2||x + 2|.

Next, notice that in the interval 1 < x < 3, |x + 2| < 5. Because of this, if ε ≥ 5, we can simply choose δ = 1, which yields |x - 2| < 1. This, in conjunction with the fact that |x + 2| < 5 (remember, that when δ = 1, we are working on the interval (1, 3) on the x-axis) implies that |x + 2||x - 2| < |x + 2| < 5 ≤ ε.

Thus, if ε ≥ 5, then choosing δ = 1, yields¹

Next, we have to find an appropriate δ for ε < 5. In this case, ε/5 < 1. When we choose δ = ε/5 (since ε/5 < 1, the equation |x + 2| < 5 still applies), we get |x - 2| < ε/5. This means that |x - 2||x + 2| < (ε/5)(5) = ε. Thus, we have

Note that when ε = 5, both of the above two arguments apply, since δ = ε/5 = 1. From the above arguments, we conclude that whenever δ equals the minimum of 1 and ε/5 (in other words, whichever one is lowest), then the statement

is true. Since we have succeeded in finding a δ > 0 for each ε > 0 and for which the above statement is true, then we have also succeeded in proving the statement

■

Note that in the above example, it was rather cumbersome and not at all obvious how to initially go about proving the limit. It is even more difficult to use the ε-δ definition to find an unknown limit. Because of that, the  ε-δ definition is almost never used to find limits. Instead, more sophisticated techniques have been developed to determine limits, which will shortly be presented on this blog. Instead, the  ε-δ definition is really useful for proving properties of limits, which we will also get to shortly and whether specific types of limits exist. For now, I would like to present some limits whose odd behaviors prevent the limits from actually existing.

Differing Behavior From the Left and from the Right


Consider the function y = |x|/x. The plot of this function is shown in Figure 1.1

Figure 1.1

As you can see from Figure 1.1, ƒ(x) is -1 if x is negative, +1 if x is positive, or undefined if x = 0. The interesting thing here is obviously the plot differs from the left and the right of x = 0. Because the function gets arbitrarily close to -1 as x approaches 0 from the left, and arbitrarily close to +1 as x approaches 0 from the right, the limit, lim_x→0 (|x|/x) does not exist (as we shall see though, there are one-sided limits which do exist for this point on the function). Formally, for any open interval (-δ, δ) on the x-axis, there will always be values of x inside that interval for which ƒ(x) = 1 and values of x for which ƒ(x) = -1.

Unbounded Behavior

The plot for y = 1/x² is shown in Figure 1.2 below:

Figure 1.2

As x approaches 0, for this function, ƒ(x) increases without bound. The smaller the interval (-δ, δ), the bigger ƒ(x) must be within that interval. Since ƒ(x does not approach a real number as x approaches 0 (∞ is not a number), we say the limit lim_x→0 1/x² does not exist. In this particular case, however, we may write

This is not a statement of the existence of the limit, but rather a statement that the function increases without bound as x approaches 0. A limit must be equal to a finite number in order to be considered as existing.

Oscillating Behavior


There are many interesting functions for which limits at specific points (or all points) do not exist for various reasons. One final important behavior that prevents limits from existing are functions that have an infinite number of oscillations between two fixed values as x approaches x₀. An example of such a function is ƒ(x) = sin 1/x. Consider the following points of the function:

ƒ(2/π) = 1
ƒ(2/3π) = -1
ƒ(2/5π) = 1
ƒ(2/7π) = -1
ƒ(2/9π) = 1
etc...

In fact, there are an infinite number of oscillations between 1 and -1 on the y-axis for any open interval (-δ, δ) about x = 0. Because of this, the function never settles down to approach any given value no matter how small of an interval you consider. See Figure 1.3 below for a plot of the function.

Figure 1.3²

Footnotes

¹ The double-struck arrow in the notation below this line means "implies," and the upside-down A can be read as "for all," so, in this instance, the statement above reads "0 < |x - 2| < 1 implies (that) |x² - 4| < ε for all ε ≥ 5." Some authors use a single-struck, or normal arrow such as "→" to mean "implies." A left-pointing arrow can be read as "is implied by" indicating that the condition to the right of the arrow implies the condition to the left, rather than the other way around, and an arrow pointing both ways is used as shorthand for "iff," which itself is shorthand for "if and only if," which means that the right condition both implies the left condition and is implied by the left condition (in other words, if one of the two conditions is true, then they are both true).

² The title of Figure 1.3 should read, "ƒ(x) = sin(1/x)" and not "ƒ(x) = sin(x)"