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 value as you approach a given value from both the left and right.  For example, consider shown in Figure 1.2 again (shown again below), but this time let’s find:

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

In the previous section I mentioned approaching the given 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 we get close to as we approach 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 from the left side. This is denoted like this:

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

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

Therefore, we can say that

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

Approaching from the right side

Similarly, if we only consider what value we approach as we get close to 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 value from one side:

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

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

Putting them together

Now, we have found both one sided limits of this function around . As we approach from the left side, we get closer to , but as we approach from the right side, we get closer to . Since we get infinitely close to two different values depending on whether we approach 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!