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:

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

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 \(y=2\).

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:

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!