One-sided Limits

To end the last lesson, Limits – Intro, I mentioned that a limit would not exist if you did not approach the same y value as you approach a given x value from both the left and right.  For example, consider f(x) shown in Figure 1.2 again (shown again below), but this time let’s find:

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$$\lim_{x \to -1}f(x).$$

In the previous section I mentioned approaching the given x value from the left and the right side to find the limit. This is necessary when finding two-sided limits (usually just referred to as “limits”), but let’s consider each side separately. In other words, we will consider what y we get close to as we approach x=-1 from the left side as one problem and from the right side as a separate problem.

Approaching from the left side

First we will approach x=-1 from the left side. This is denoted like this:

$$\lim_{x \to -1 ^{-}}f(x).$$

Notice the little ^- to the right of the -1. This tells us the we are approaching x=-1 only from the negative side, or the left side. As shown below in Figure 1.3, as we approach x=-1 from the left side, we get closer to y=2.

Figure 1.3

Therefore, we can say that

$$\lim_{x \to -1 ^{-}}f(x) = 2.$$

Approaching from the right side

Similarly, if we only consider what y value we approach as we get close to x=-1 from the right side, or the positive side, we are finding:

$$\lim_{x \to -1 ^{+}}f(x).$$

Just like before, we only want to consider approaching this specific x value from one side:

right sided limit
Figure 1.4

As you can see, if we start on this function to the right of x=-1 and we move toward x=-1 along the function, we get closer and closer to a y value of 4. Therefore, we can say that this one-sided limit has a value of 4, or

$$\lim_{x \to -1 ^{+}}f(x) = 4.$$

Putting them together

Now, we have found both one sided limits of this function around x=-1. As we approach x=-1 from the left side, we get closer to y=2, but as we approach x=-1 from the right side, we get closer to y=4. Since we get infinitely close to two different y values depending on whether we approach x=-1 from the left side versus the right side, this two-sided limit actually does not exist. So,

$$\lim_{x \to -1}f(x) \ \ DNE.$$

Why do we need to consider each one sided limit separately?

This is an important thing to remember. In order to find any two-sided limit, you will instead find each one sided limit. If both one-sided limits are the same, then the two-sided limit will also be that same value. However, if the one-sided limits are different, the two-sided limit does not exist. In other words, $$if \ \lim_{x \to a ^{-}}f(x) = \lim_{x \to a ^{+}}f(x) = b, \ then \ \lim_{x \to a}f(x) = b.$$ $$And \ if \ \lim_{x \to a ^{-}}f(x) \neq \lim_{x \to a ^{+}}f(x), \ then \ \lim_{x \to a}f(x) \ does \ not \ exist.$$

Another method that can be used to evaluate one-sided limits if you don’t have a graph of the function available is using the properties of limits. You can learn more about the limit properties here.

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For more on limits go check out my limits page.  There’s a list of lessons and practice problems there all about limits.  Take a look to get some more practice with limits.  If you have a specific topic or problem you’re looking for and can’t find it, then email me at jakesmathlessons@gmail.com.  Send me your questions and I’ll be sure to point you in the right direction to get them answered.  I may even write a lesson and post it to make sure your question gets answered!

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