Welcome back, everyone. Here, we're going to be taking our very first look at limits. Now the first time that you come across a limit, they may seem a bit intimidating. But in finding a limit, all we're doing is looking at what a function is doing around some given value of x. Now we can do this easily by either looking at a graph or by creating a table of values. Now here I'm going to show you both of these ways of finding limits and we'll also walk through some of the notation that you'll encounter when dealing with limits. So let's go ahead and jump in here.

Now let's first take a look at our limit notation. This is what you'll see anytime you're working with a limit and we can read this as the limit of f of x, our function, as x approaches c, the value for which we're taking the limit. Now looking at this limit here, this is asking us to find the limit of x squared, that's our function f of x, as x approaches 2. That's our c value. Now our limit will always be equal to some value n, so let's actually find that value for this limit here.

Now remember that I said in finding a limit, we want to look at what our function is doing around that given value of x. Now we can get a bit more specific here and say that a limit is actually the y value that a function f of x goes to as x gets really close to a given value. So here, we're looking for the y value that our function x2 goes to as x gets really close to 2. Now what does that mean x gets really close to 2? Well, it means that we don't care what's happening when x is 2 but just when x gets really, really close to 2. So looking at our graph here, as x gets really, really close to 2 from either side, we want to look at what y value our function is going to.

Now first looking at x from that left side closing in on 2, getting really, really close to it, I see that my function seems to be approaching a y value of 4. Now I can also think about this numerically and actually plug values in that are getting really, really close to 2 into my function. So I can take numbers like 1.99 or closing in on 2 even further, 1.999. Now actually plugging these values into my function squaring them, here I get 3.96 and 3.996. Now here again, I can see that these values seem to be getting close to a y value of 4. Now let's look at that other side. Now closing in on 2, getting really, really close to it coming in from that right side, I see again that my function appears to be approaching or going to a value of 4. Now, again, I can look at this numerically and actually plug values into my function that are closing in on 2 from that other side, like 2.01 and 2.001. Now again, plugging these values into my function, here I get 4.04 and 4.004. So again, from this side, I see that we seem to be approaching a y value of 4.

Now again, here it does not matter what's happening when x is 2, but just as x gets really, really close to 2. Now here from both our graph and from our table of values, we see that the y value our function is going to as x gets really close to 2 is 4. So our limit of x2 as x approaches 2 is 4. Now here we used 2 different ways of finding this limit. We've used a table and we used a graph. Now often you'll just be using one of these methods and problems will actually specify which method they want you to use. So let's work through a couple more examples together here.

Now this first problem asks us to find the limit of x+4 as x approaches 1 by creating a table of values. Now we want to look at values really, really close to 1 here. So let's first look at values closing in on 1 from this left side. So these values are at less than 1, but getting really close to it. So here, I want to look at values like 0.99 and 0.999. Now plugging these values into my function x+4, here I get 4.99 and 4.999. So I can already see that this function seems to be going to a y value of 5. But let's also look at values really, really close to 1 closing in on it from the other side. So here values like 1.01 and 1.001, getting even closer there. Again, plugging these values into my function, here I get 5.01 and 5.001. So here it is clear that we are going to a y value of 5. So the limit of x+4 as x approaches 1 is 5, and that's my answer here.

Now something that you may have noticed for this problem as well as the one that we worked out above is that I could have just plugged my c value into my function. So here, if I would have just plugged 1 into my function x+4, I would have gotten 5, which is exactly what our limit is. So why am I using this language of x getting really, really close to 1 if I could have just plugged 1 in from the beginning and got my answer? Well, the reason is that this is not always going to be true. You can't always just rely on plugging in that function value because your limit will not always be equal to the function value. So we can't just plug c in from the beginning. So with that in mind, let's look at this final example here.

We want to find the limit of f of x as x approaches 3 using this graph here. Now looking at our graph, we want to look at x getting really, really close to 3 from either side. Now see here looking at this graph coming in from this left side, I see as x gets close to 3, I'm approaching a y value of 1. Then coming in from the other side I see the same thing happening, as x gets really really close to 3 our function is going to a y value of 1. So the limit of f of x as x approaches 3 is 1. And here, if I would have just plugged 3 into my function or just looked at that function value on my graph, I would have gotten 4, which is a completely wrong answer. That is not what my limit is because my limit is 1. So here it's very clear that our limit will indeed not always be equal to our function value. We need to look at x getting really, really close to c, not x actually being equal to c because we never know if our function value is going to be something entirely different like we saw on our graph here or if it even exists at that point. And it doesn't matter because we only care what's going on around it. So with this in mind, let's work through some more practice together. Thanks for watching, and let me know if you have questions.