When finding a limit, we look for the y value that our function is going to as x gets really, really close to a given value from either side. Now, in looking at what our function is doing from either side of this x value, we've actually been finding what's called one-sided limits, which is something that you may be explicitly asked to find. Now, because we've already been doing this, just not calling it one-sided limits, the only thing that's really new here is the notation. So let's go ahead and jump right in.
In working with a one-sided limit, we still write the limit of f(x), our function. But instead of x just approaching c, we have x approaching c with a little negative sign. And we read this as the limit of f(x) as x approaches c from the left. That's our left-sided limit. Now, for our right-sided limit, we have the limit of f(x) as x approaches c with a little positive sign from the right. To remember which sign represents which limit, think about your coordinate system because we know that our x values from the left are negative, and our x values going to the right are positive. Now that we've seen this notation, let's take a look at this function here and actually find some limits.
Let's start here with our left-sided limit. So, the limit of f(x) as x approaches 3 from the left. First, looking at my graph here, as we get really, really close to 3 coming in from that left side, I see that my function is approaching a y value of 1. I can plug values getting really, really close to 3 from the left side into my function. Here as we're closing in on 3, I use values like 2.992, 2.999. Now, because this is a piecewise function and these values are less than 3, my function here is equal to x minus 2. So plugging these values into my function, I get outputs like 0.992, 0.999. Just like I saw on my graph, based on these values, my function is going to a y value of 1. So here, my left-sided limit, the limit of f(x) as x approaches 3 from the left, is 1.
Now, let's take a look at our right-sided limit here, the limit of f(x) as x approaches 3 from the right. Looking at our graph, as we get really, really close to 3 coming in from that right side, I see that my function is going to a y value of 4. I look at those values getting really, really close to 3 from that right side, at 3.001, 3.013. Again, plugging those into my function, which here since these values are greater than or equal to 3, my function is equal to 4. Based on these values, just like I saw on my graph, I see that my function is going to a y value of 4. So, my right-sided limit, the limit of f(x) as x approaches 3 from the right, is equal to 4.
Now, what if I asked you to find the limit of f(x) as x approaches 3? Not from the left or from the right, but just as x approaches 3. This is a bit confusing because our one-sided limits show different outcomes. As x gets really, really close to 3 from either side, we see different things happening. Because this function does not approach the same value from both sides, the limit does not exist or is abbreviated as DNE. Here, with this function, as x approaches 3 from either side, going to a y value of 1 from one side and to a y value of 4 from the other, the limit does not exist. Here we were able to find our one-sided limits, and because these one-sided limits were not equal, that told us that our limit does not exist.
Let's take a look at another example. We use this graph to find the given limits. Now, the first limit we're asked to find here is the limit of f(x) as x approaches 1 from the left. As x approaches 1 coming in from that left side, I see that my function is going to a y value of -1. So here, the limit of f(x) as x approaches 1 from the left, my left-sided limit, is -1.
Now, moving on to our next limit. Here we have our right-sided limit, the limit of f(x) as x approaches 1 from the right. Again, taking a look at our graph, as x gets really close to 1 coming in from that right side, I see again that my function is going to a y value of -1. So here, my right-sided limit is the same as my left-sided limit, -1.
Now, let's look at one final limit, the limit of f(x) as x approaches 1. We want to pay attention to what's happening as x approaches 1 from either side, but we have already done that by finding our one-sided limits. Here, since our one-sided limits are equal, they are both -1, that tells us that our limit here, the limit of f(x) as x approaches 1, is also -1. So, when working with one-sided limits, if they are not the same, then your limit does not exist. But if they are the same, then your limit does exist.
Now that we've seen and put a notation to these one-sided limits and know how to use them to determine whether a limit exists, let's continue practicing. Thanks for watching, and I'll see you in the next one.