So let us try to generalize the concept of the limit of the sequence into the case of functions of real domain. Basically, what we've actually already have, we have the idea that the limit is a value of not just any real value which resembles our functions or our sequence the most. The rule here is that if you have any given deviation, then we can assume that our sequence or our function does not deviate much from the limit if it get close enough to the limit point. So let us try to formally write this down. Well, this is nightmarish because we are going to use our quantifiers as been transduced on the previous week but everything will be fine. So let us start from our picture. What we do have here? We do have here our function, this is a blue curve and then have a point of interest or limit point a here. So what's an idea? What I'm going to say is that this function have a limit equal to c as x approaches a and c is drawing over there. So what does it mean? It basically means that if we just state that we allow our function to deviate from the limit by Epsilon, the Epsilon just any possible real positive value, then we create some boundaries on our functions were derived, c plus Epsilon and c minus epsilon. So this is our epsilon tube here and the idea is that we can get as close to point a as the function doesn't break our Epsilon tube. In our case, we've come up with the idea of Delta here, neighborhood of the point a that's Epsilon tube and Delta tube intersects in rectangle and our function wholly lies within the rectangle, that nice. So the second thing we need to do is to change our Epsilon because what we want achieve we want to achieve the ideas that we can do this procedure for all possible deviations so that's how our definition rock. It's extremely formal one but we should always start with the formal things. So for all deviations, we can get as close to our limits point thus function does not deviate from its linked much than previously stated division. That's how it basically can be read in natural language, that's fine. But there is bomer here, why do we need to write that our absolute value of x minus a or this the distance between arbitrary point in the neighborhood of the a is smaller than Delta which basically means at its neighborhood. But what is greater than 0, greater than 0 actually means that we are not considering as appoint a itself. So basically, it means that we are looking at every where there is a function as neighborhood except the point itself and twice it. Let us assume that there is some enemy around us and this enemy come at the very point a in just damaged our functions and served that f from a for example is not how nice easy blue curve and sample as a point for example this one. Remember our definition, well hand-waving definition so what we've said previously, we've said that the limit is the value that resembles our function in the neighborhood. As we get as close to the limit point, we are not supposed to be ends on this point. We are considering only functions in the neighborhood of it and the neighborhood function doesn't actually care about the value of the function at the very point itself so we should actually expel our x equal to a from zircons duration of the limit, so it's tricky. Sometimes people will use those ideas that we need to have an as more sequence-based definition for the limit function and it's expectable because it does write in the following manner. If we assume any sequence of arguments of our function x_n which approach is our limit point n which can be done because a is our limit point, then the sequence of function values on this sequence of functions hiker events should necessarily approach c, approach it's limit. It nice because it operates in the same joint set that we've already have been established as a limit of a sequence but the problem here is that we're talking about all possible approaches to the point a, all possible sequences that approaches point a which nightmarish object. You can even just imagine how much sequences approaches point a. So the first definition is usually used to prove that there is a limit and to calculate it and the second definition is used to prove that there is no limits or this point is not really much. So it used to prove an absence of a limit.
