When finding a limit, we look for the y value that our function is going to as x gets 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 asked explicitly 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 notation. So let's go ahead and jump right in. Now in working with a one-sided limit, we still write the limit of f of x, our function.
But instead of x just approaching c, we have x approaching c with this little negative sign. And we read this as the limit of f of 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 of x as x approaches c with this little positive sign from the right. Now in order to remember which sign represents which limit, you can 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.
So now that we've seen this notation, let's take a look at this function here and actually find some limits. Now let's start here with our left sided limit. So limit of f of x as x approaches 3 from the left. That's what that little negative sign tells me. Now 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.
Now I can actually plug values that are getting really close to 3 from that left side into my function. And here as we're closing in on 3, I want to look at values like 2.99, 2.999. Now because this is a piecewise function and these values are less than 3, I know that my function here is equal to x minus 2. So plugging these values into my function, here I get 0.99 and 0.999. So just like I saw on my graph, I can see here based on these values that my function is going to a y value of 1.
So here, my left sided limit, limx→3−f(x)=1. Now let's take a look at our right sided limit here, the limit of f of x as x approaches 3 from the right. Now looking at our graph first here, 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. And again, I can look at those values that are getting really, really close to 3 from that right side closing in on it, 3.01, 3.001, again, plugging those into my function, which here, since these values are greater than or equal to 3, my function here is just equal to 4. So again, based on these values, just like I saw on my graph, I can see that my function is going to a y value of 4.
So my right sided limit, limx→3+f(x)=4. So here we found our one-sided limits from the left and from the right. But what if I now asked you to find the limit of f of x as x approaches 3? Not from the left or from the right, but just as x approaches 3. Well, this might be a bit confusing because we just saw based on our one-sided limit that as x gets really, really close to 3 from either side, we see some different things happening.
And this tells us that because this function does not approach the same value from both sides, the limit actually does not exist or abbreviated DNE. Now this is something that you may have heard before. The limit does not exist. And here with this function, because as x approaches 3 from either side, we see some different things happening from one side going to a y value of 1 and from the other going to a y value of 4. This limit, does not exist.
Now here, we were able to find our one-sided limits. And because these one-sided limits were not equal to each other, that told us that our limit does not exist. So while you may be asked to explicitly find your one-sided limits, you may also be asked to use those one-sided limits to determine whether or not your regular limit, not your one-sided limit, exists or not. So let's go ahead and take a look at another example here. We want to use this graph to find the given limits.
Now the first limit that we're asked to find here is the limit of f of x as x approaches 1 from the left. That's what remember that that's what my little negative sign tells me here. So looking at my graph here, as x approaches 1 coming in from that left side, I see that my function is going to a y value of negative one. So here, my limit of f of x as x approaches 1 from the left, my left sided limit is negative one. Now let's move on to our next limit.
Here we have our right sided limit, the limit of f of x as x approaches 1 from the right. Now again, taking a look at our graph here, as x gets really, really close to 1 coming in from that right side, I see again that my function is going to a y value of negative 1. So here, my right sided limit is going to be the exact same as my left sided limit, negative one. Now let's look at one final limit here, the limit of f of x as x approaches 1. Just 1, not from the left or from the right.
So that means that we want to pay attention to what's happening as x approaches 1 from either side. But we already did that by finding our one-sided limits. Now here, since our one-sided limits are equal to each other, they're both negative one, that tells us that our limit here, the limit of f of x as x approaches 1, is also negative one, and we're done here. 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.
So now that we've seen and put a notation to these one-sided limits and know how to use them to determine whether or not a limit exists. Let's continue practicing. Thanks for watching, and I'll see you in the next one.
