Variable approaching infinity - what happens

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

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English [Auto] So another way to write it would be like this limit. As x approaches 1 x square minus one X minus one equals two so it's profoundly different from what does this function equal to when X equals 1. No this sentence or that or this mathematical language if you translated into English means what does this function approach to from both sides from left and right. If that if x approaches to one from both sides what does this function approach do and this function approaches to 2. OK. Now sometimes you can have a situation where where for one side you approach one value and from another side you approach another value. For example if I have this kind of function X and Y and then I just have a function like this and like this and this is mine is. And this is plus 1. So the function here would be defined like this. So it would be minus 1 if X is smaller than zero. And remember if it's just smaller and that's more or equal to but just smaller than zero is not included. But that one is when X is bigger or equal to zero. So now because of this size zero is included. So I have a void here and I have a point here. Now of course here you can see that the that you approach different values as you approach X equal as you approach x 0 you approach different values from the point of view of Y. So the way I would write it is like this limit x x x approaches to zero from the negative side which is from this side from the negative side. Then why X would approach minus 1. Right. So from the negative side you approach X equals zero and then you approach Y minus one. However if you approach x if x approaches zero from the positive side. So you put a plus here if you approach 0 if x approaches zero from the positive side then why would be one. So these are called one side limits because on one side you approach one value and then from the other side you would approach an undervalued. Now I have an exercise for you. OK. Just pause the video and try to solve it OK. Well if x approaches infinity which means it goes infinitely and infinite distance in the direction of x Well then you can see that the function would approach zero. It would never touch zero. It would never touch y equals zero but it would approach with zero. So that's why when ex-school zero You cannot have a value for Y because then it's undefined. However as you as x approaches infinity plus infinity why would approach zero. Now if if x approaches minus infinity which is in this direction then why would also approach zero. So the further I go the closer this line gets to zero that would be zero. If x approaches zero from the positive side. So from this side then this value here would never touch the y axis but it would go further and further up because I would divide by a smaller and smaller number. So I would end up with plus infinity. So the correct way of saying is is that as x approaches zero from the positive side y approaches plus infinity and if x approaches zero from the negative side then why would approach. Mine is infinity and you could see from the graph and also you would divide by smaller and smaller number. And just to make it a little bit more intuitive like what does it mean that it never touches y equals zero. Well let's say that if I have you 5 x here and Y here then let's see if I have one here. So that's my first interval but let's say for the second interval I take half of it. Right. And then I would have zero point five for the third interval. I would take another half. So it would have 0.25 for the fourth interval. I would have I would take another half which would be zero point zero or six to five. And I can go on and on and on and I can always take half of the previous value which means that I will never quite reach zero. But if X goes to infinity as x approaches infinity the further and further I go the closer and closer y approaches to zero. So that's the intuition behind it. I also want to show you very quickly a very special function in mathematics which is this one one plus one over X and then all that to the power of x. The reason why it's so important in mathematics is because if X approach to infinity and by the way this is not an exercise and just showing it to as x approaches infinity your function would be something like this. And he would approach a certain value. And the value that it will approach is the Ehlers number. So Ehlers number which is 2 points seven is also an irrational number. It has an infinite amount of decimals but this is where it comes from. From this function so as x approaches infinity this term becomes smaller but then you have this term in the power which becomes bigger and through a mathematical proof. It has been proven in mathematics that you will achieve Elorn number.