Intro to Limits | Calculus - Nerdstudy

So your Calculus journey begins.

What is a limit? How do you find the limit? What does this "limit" thing really mean anyways?

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So far in mathematics, we are used to listening to questions such as, when 'x' equals 1, what is 'y'? So for something like that we know right away how to answer this. We look at 1 for 'x' and we see that 'y' is 2. That was a graphical way to do it, the equation way to do that would be to simply substitute 1 for 'x' and we would get 1 + 1, for this equation, equals 2. Now, the study of Calculus pertains to this word called "limits," and limits would be the introduction to calculus. Now limits are all about the idea of "Approaching." We're not going to look at what 'y' is when 'x' is 1. We're now going to look at what 'y' seems to be approaching when 'x' is approaching 1. So one way for us to understand this is by taking a look at the graph and maybe making a table of information, so that we can think about what it would mean to approach a number. So since our equation is x +1, we can actually take a look at what an 'x' of 0.8 would give us. And if we put 0.8 + 1, we would actually get 1.8. So what we did here is... I just started off with a number that's fairly close to 1. And it was just a random arbitrary number, but the idea of approaching would not really be fulfilled unless we do, at least, more than 1 set of values. 0.8 might be near 1, but we can now say that we are approaching 1 when we get closer to 1 from 0.8. So let's try, maybe, 0.999. This is a lot closer to 1. And we see that the 'y' value is now 1.999. And we can see that it seems like as 'x' is approaching 1, 'y' seems to be approaching a number closer to two. Now, if you thought for a moment that we are not approaching a number closer to 2, all you would have to do is to try another 'x' that is even closer to 1. You could actually repeat this many many times, but of course we're not going to do it too many times in this lesson, but the idea is this: If you're getting closer and closer to 1 for 'x' and for your 'y' it seems to be getting closer and closer to a number, in this case it becomes extremely obvious that it seems to be getting closer and closer to 2. Now this is an interesting thing that we want to take a look at before we move on. We we just did was this: we actually got closer to 1 from the left side. So we get closer to 1 from numbers lower than 1. The idea of getting closer to a number can easily happen from the opposite direction, so numbers greater than 1 getting close to 1. So instead of going this way, we could have also looked at the idea of closer going this way. So if you took a number, for example, 1.15 that is fairly close to 1, you would have gotten a 'y' value of 2.15. Now again, we want to think about when 'x' is approaching 1, what is 'y' approaching. So what we do is we get closer to 1 from 1.15, which would be let's say 1.00001. And we would get a 'y' value of 2.0001. So if you look at what happened to our 'y', it's pretty close to 2 all of a sudden. And you know right away that if you get even closer to 1 for 'x' you would probably get even closer to 2 for 'y.' So, by the use of this table of values, and also just by visually looking at the graph that we have, we can see then that every single time we take an 'x,' that is close to 1 and then even closer to 1, it seems like we are certainly getting closer and closer to 2. Now the problem with this example, and maybe we shouldn't even say that it's a problem, but this example doesn't actually help us to distinguish the difference between when x is equal to 1, and when x is approaching 1. The first question is really asking us, when 'x' is equal to 1, what is 'y?' So we're looking at no other points but exactly when x is exactly that one number, which is 1. And we know exactly that 'y' is equal to 2. The section question is really asking us, when 'x' is approaching 1... so basically the idea of approaching would be: every number near 1, getting closer and closer to 1 but not 1 itself ... in this case, what is 'y' approaching? The second question is one where we will always ignore this table of values.