 Another important topic in limits concerns what are called one-sided limits, and this occurs in a number of different cases where we might only be able to approach a given value from one direction or another. So we might have a situation where we can only approach a given value of x from above. So let's say we want to know what happens to f of x as x approaches an a value from above, or we also would say that this is from the right because we are looking at the approach to the number from the right side, if you think about a number line, and we'd indicate this limit as x approaches a from above, there's a superscripted plus there, of our f of x, and we might consider what that limit is. Or we might also consider what happens as x approaches a from below, and again if you think about this on the number line, we might also be looking at that x approaching a from the left, and we write it this way, the limit as x approaches a superscripted minus from below of f of x. Now it's an important point to make, which is that these are one-sided limits, we're either approaching a from above, or we're approaching a from below, but if we want to take a look at the limit as x approaches a, we have an unqualified limit, and we need to have a common value of the two limits, but if those two limits are not equal, then the limit itself does not exist. So if our limit as x approaches a from above, and our limit as x approaches a from below, if they are equal, our unqualified limit is the common value, otherwise we have no limit. For example, let's talk about the limit as x gets close to four from above of the square root of x minus four, and while we're at it, let's see if we can defend our answer numerically. So here we're approaching four from above, and so we want to consider what happens to square root of x minus four for x values that are getting close to four, but always staying above four. Now if you actually look at the function, you'll see why we can't actually approach four from below. If x is less than four, we're taking the square root of a negative number, which for real variable calculus is undefined. Now we want to defend our answer numerically, and it might not be a bad idea. Let's actually start with a numerical approach. So we'll consider values of x that are close to four, always staying a little bit more than four, and what the corresponding value of the square root of x minus four is. So I might try 4.01, substituting that in, and I get that 0.3162 as the approximate value. Or I might get a little closer to four, try 4.01, again still a little bit more than four, and I'll substitute that in, and after all the dust clears, 0.1 as my value of the function, again trying a closer value, 4.001, and binding my value 0.0316. So if x is close to four, but always staying a little bit more than four, then here are my function values. Now it might not be entirely clear what's happening to the function at that point, so let's try an algebraic approach. As x gets close to four from above, square root x minus four gets close to square root of four minus four, which is square root of zero, or just zero. And that suggests that as x gets close to four from above, x square root of four, square root of x minus four is zero, and our table tends to support that. For x values that are close to four, our values of square root of x minus four are also fairly close to zero. So if I want to give a complete answer to the question, find the limit, and also defend the answer numerically, I want to include both the table and the work that leads to the value of the limit. Now these one-sided limits are particularly useful for functions that are defined piecewise, and what that means is why I have a function f of x equals, well, sometimes it's going to use one particular rule, and sometimes in another piece it'll use a different rule. So in this case my function uses the rule 3x minus two, if x is greater than or equal to four, and if it's going to use the rule x squared minus six, if x is less than four, and I can still try to find the limit as x gets close to four of my function. Now notice that the rule changes at four. There's a break at x equal to four, where we switch from one rule to the other, and since that's where we're trying to find the limit as x approaches four, then what we will have to do is we should take a look at the one-sided limits as x gets close to four. So we'll start off by finding the limit as x gets close to four, but always staying above it to f of x. Now x is getting too close to four from above, which means that our x values are always going to be greater than four, which means that my function, if x is greater than four, I'm going to use this top line here. So my function, for the purposes of finding this limit, is going to look like 3x minus two. So I'll include that in as my actual function value, and now I can apply the same logic as before. As x gets close to four, but always staying above it, this expression 3x minus two looks like it's getting close to three times four minus two. Looks like it's getting close to ten. Likewise, I can also take a look at the limit as x gets close to four, but always staying a little bit below it. And if I'm a little bit below four, if I'm less than four, then my function is x squared minus six. So I am going to look at the limit as x approaches four from below of x squared minus six, and as x gets close to four, my expression gets close to four squared minus six, also equal to ten. And the important observation to make, the one-sided limits have the same value. The limit as x approaches four from above is ten. The limit as x approaches four from below also ten. So the unqualified limit exists and is equal to the common value. And again, just as a note on syntax, the proper and complete answer to the question is going to be the portion shown in green. We want to show what the limit as we approach x equal to four from above, the limit as we approach x equal to four from below, and make an observation that the limits agree. So the limit without plus without minus, that limit is going to be equal to the common value. We'll take a look at another one. So again, I'm going to define this function piecewise, and f of x has three different formulas, depending on whether x is greater than two, equal to two, or less than two. And I'm interested in finding the limit as x approaches two of my function. So again, my function breaks at two, and I'm interested in finding the limit at two, which means I'm going to have to find the one-sided limits. So let's start off with a wrong way of doing this. So we might just plug in x equals two into the function at x equals two. The line says f of x is equal to seven. So the wrong answer is that as x gets close to two, my function value gets close to seven. And this is wrong for several reasons. And the key reason is that the limit as x approaches two is not what actually happens at x equals two. But again, what we're looking at is what happens to f of x as x gets close to two. So let's try to do this correctly. So the function breaks at two. So we'll find the one-sided limits. So we'll start off with the limit as x gets close to two, but always staying a little bit above two. And my function, if x is a little bit more than two, looks like two x minus five. So that limit is going to be the limit as x approaches two of two x minus five. And this expression gets close to the value two times two minus five gets close to the value negative one. Likewise, I can take a look at the limit as x gets close to two, this time staying a little bit below two. And again, if x is less than two, my function looks like one minus x. So I'm looking at the limit as x approaches two from below of one minus x. And that looks like it's going to head towards one minus two. Looks like it's heading towards negative one. And now the limit from above, the limit from below, have the same value negative one. So the unqualified limit is going to be the common value. Now note that in this particular case, we didn't have to find a numerical support for it. The question didn't ask us to find the numerical support for it. But it did ask to find the limit as x approaches two. And it's important to note that in this case, the complete answer is going to be the portion that is shown in green. Finding the limit as we approach from above, the limit as we approach from below, and the observation that the two limits are the same. And so the limit itself is going to be the common value. So we'll take a look at one more example here. So this time we have yet another piecewise function. And again, here our break is at x equal to four. And we want to find the limit as x gets close to four of our function. So we'll find our one-sided limits starting with as x gets close to four. But saying above it, if x is bigger than four, I'm going to use this line here, x squared as my function expression. And as x gets close to four, that gets close to four squared or 16. I'll look at the limit as x approaches four from below. I'm a little bit less than four. And that means I'm going to use the top line here. My function expression is going to be 2x. And as x gets close to four, that looks like it's getting close to eight. And we have a problem. The limit from above is 16. The limit from below is eight. And the one-sided limits do not agree, which means that there is no unqualified limit. So we do want to say something to that effect since the limits are not equal, then the limit itself does not exist. And again, the complete answer to this question is going to be shown in green. Our limit from above, our limit from below, and some observation that because the limits are different, the unqualified limit does not exist.