Variable equaling VS approaching something - what's the difference

Mark Misin
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:36 of on-demand video • Updated October 2020

  • You will develop very strong intuition & understanding in Calculus
  • You will learn how to apply Calculus in real life to the level not seen in other courses
English [Auto] We'll come back so starting from this lecture we're going to start examining how non-straight line functions change. In other words we're going to start looking at something called derivatives. But before we go into that I need to make sure that you understand the concept of limits. And I really wanted to have an intuition for that. And in order to do that I am giving you four functions. Let's look at that. So I have a straight line function here h as a function of x equals Expo's 1 OK it's just a straight line. Then I also have G as a function of x. Now it's also a straight line function but it is different. It has a denominator so it it has components that would cancel out. However as you know you can divide by zero. So if x equals 1 then you would have zero in the denominator which you can have. So you can have a value here at the ECC equals 1. But then I define this function in a piecewise way and I just say that OK when X equals 1 then my function equals 1 and I can do that. I can define functions the way I want. Now this other function f as a function of x is the same function like this one. But here I don't do anything about X equals 1 which means that since I can divide by zero when X equals 1 I cannot have a value here. So I have a straight line function. But then at X equals 1. I don't have a value in this case. I have a straight line function that follows this line but at one I don't have a value here. I have a value here at X equals 1. And then I also have us a parabolic function that is shifted one unit to the right and two units up. And I've defined it like this when x is not 1. Then it follows this rule but when X equals 1 then my function equals 1. And I can do that because again I can define functions the way I want. However I want you to look at those two sentences. And if you could just take your time to read them through and think about them they are profoundly very different sentences two very different sentences. The first sentence is what does a function equal to an X equals one OK. Well in this case the sentence essentially tells us to take X equals 1 and put it into our equations right. And we know that from this function if X equals 1 then then we have h equals to write X equal one equals two. Now in this function we know that X equals 1 then g of X equals 1 as well from this function if X equals 1. It's undefined because if equals 1 denominator because it becomes 0 and then you can divide by zero. So you can have a value here and then with the parabolic function if X equals 1 then according to this function you have Y which is 1. So that is the first sentence but the second sentence is completely different if you think about it. What does a function approach do as x approaches one from both sides. Now what does that mean. Well it means exactly that. What does this function approach to when x approaches one from both sides. So if I take this approach X and this function one from this side and from this side from both sides why then what do I approach. What does g approach Well in fact if you look at it then you can see that as you get closer and closer to execute one does this function approach one well not really because if you approach X let's say from the left side then this function doesn't approach one it approaches to right even though G when X equals one G is one. But as I get closer and closer to two Ezekias one I don't get closer and closer to 2 1 I get closer and closer to 2 right and the same thing here if I get closer and closer to exit cause one from this side from the positive side then I get closer and closer to G close to and the same thing is here if I get closer and closer to actually calls 1 I get closer and closer to a sequence to write the same thing here. If I get closer to sequel's one from both sides even though I don't have a value here. But as I get closer to 1 I approach two and the same thing here I get closer and closer to X-C cause one and even though y equals 1 at the X equals one. But it doesn't mean that I approach one approach to red. Does that make sense if you look at it. So it's one thing to say that OK what does a function equal to equal something. And another thing is to say what do you approach when x approaches 1. So these are two different things. And from here you can see that when H when xcuse 1 then H equals two equals one white is 1 and f is undefined. But all those four functions all those four functions as x approaches one all those four functions they approach to. So these are two different things. And the second sentence is is let's say abbreviated with something called limits. So mathematically I described the first sentence like this h as a function of one equals two g is a function of one equals one y as a function of one equals one. F is a function of one equals undefined but the second sentence mathematically is written like this limit as x approaches 1 h approaches to limit x approaches 1 g approaches to limit as x approaches 1 F and then why they all approach to. So I thought of of a small example to make the concept of limits a little bit more intuitive. So of course it's not a mathematical example but just to give you this intuition this logic. And very simply put this is a road and you are in a car you're driving in this direction and starting from here you will have posts. Right you have post number one two three four five six seven and then you can measure the positions of the posts like this. Position X one y y. And this is the r coordinate system here. Let's say x y. So at position X one y one we get posts one position at position X to Y two we get pushed to the position x 4 y for we get post 4. Now as you go in you see those post from your column. Right. So you see those posts and now you see there is a river here. All right there's a river. So at present and this is position x 8 y. So if I ask your question what do you have a position and say why. Well if you put the equal sign then you have River. However from the card you don't see the river from the car you will assume that you will have post 8 because you don't see the river. However based on the previous experience that you see that OK after a certain small distance I see a new post. And this is the sixth post and then I have a I'm already seeing the seventh post. So in my view as a car driver as I approach this position ex-aide Why am I. I assume that as I approach this position I expect to see post a based on my previous experiences based on what I had seen before. So. So that's the point with limits. If you just make if you say position x 8 y equals something then you have to say river because that's the reality. However if you say that as I approach X 8 wide then based on my Based on the Hinz based on my previous experience what is it that I'm approaching too. So based on my previous experience I would say that I approach post 8. So that's what the limit can is it's like OK it has nothing to do with what equals something but rather than based on the previous experiences what value should I approach. So if I have a parabola and I don't have a value here and I have values that here. However as I go along this line based on my previous experience I should have value here even though I have the value here but it's an irregularity. The limit will tell you what value I should have based on my previous experience in my previous experience from here and here says that my values should be here. So I hope that it gave you a little bit of more intuition for limits. All right. Thank you. And see you soon.