in this channel we have talked about derivatives and integrals, very important mathematical tools within calculus and whose development has allowed the creation of new physical theories and has given us a better understanding of how the universe works and what are the laws that govern it. But there is a very important concept in fact the most important in calculus because it is considered the basis of all calculus, the limit [Music] The limit is a fundamental piece in our study of calculus because many important concepts are defined using limits, the definite integral, for example, is the limit of the sum of rhymes and the derivative is the limit of the incremental quotient. When we analyze the continuity of a function, we will also use limits, so really understanding what a limit is is something very important in our study of calculus. But before talking in more detail, let's analyze some situations to get an intuitive idea about what a limit is, think about the following idea: you can get too close to something without ever stopping getting closer and still never reach it. Yes, I know. It sounds quite strange, but imagine the following situation: you are 8 meters away from a wall and for some strange reason you decide to do the following: move towards the wall but always traveling half the distance, that is, at the beginning you are about 8 meters from the wall, therefore, in your first trip you will advance half this distance and you will be only 4 meters from the wall and then you travel half the way again and you will be only two meters from the wall and you travel half the way again, so now you are only one meter from the wall and you travel half the way, so you will be half a meter from the wall, then a quarter of a meter, then an eighth of a meter, and if you have noticed, you can continue traveling half the way indefinitely, getting closer and closer to the wall. But in what situation would you end up touching the wall? Well, when you take an infinite number of steps, that is, at the limit of infinity, we end up touching the wall. Now let's analyze another situation as interesting as the previous one, let's suppose that we have a square whose side is one unit, so the area of this square is equal to one unit squared, very well. Now we are going to divide this square in half, so the area of each of the pieces will be half a unit squared now let's divide this shaded part in half so each piece will now have an area of one-quarter units squared now let's divide this other shaded part in half so each part will now have an area of one- eighth units squared and again let's divide the shaded part in half so each part will now have an area of 16 units squared and if we keep doing this of dividing each part in half we'll get 1 over 32 then 1 over 64 then 1 over 128 and then 1,256 And if you noticed we can keep dividing each piece in half an infinite amount of times now think about this if you add up the areas of all those pieces the area must equal one unit squared which is the area of the whole square that is to say this sum with infinite terms is exactly equal to one if you were to add up a million terms the sum would be equal to a number that is very, very close to one but not exactly one therefore for it to be exactly equal to one this sum would have to have an amount infinite of terms we can represent this sum in the following way 1 divided by 2 to the power of one plus one divided by two to the square plus one divided by two to the cube plus one divided by two to the power of four plus one divided by two to the power of 5 plus ellipsis And so on using the symbol of the summation to represent it in a more compact way we can write it in the following way summation of the term 1 divided by 2 to the power of n from n equal to 1 to infinity is exactly equal to 1 and yes this is a very good example of how a sum of infinite terms can result in a finite result which is equal to 1 at the limit of infinity the first thing we must understand is the concept of Limit and what it is used for and for this let's look at an example with the function fdx is equal to x squared whose graph we can see here let's suppose that we are interested in analyzing how this function behaves as we take values closer and closer to two we know that when x is equal to 2 the function raises it to the square that is 2 squared and returns the value of 4 but we are not interested in knowing what happens when x is 2 but rather What happens when x takes values very, very close to two and for this let's start by doing the following let's start by making x equal to 1 replacing the function we obtain that F of 1 is equal to 1 squared which simply gives us 1 now let's see What happens to the function when we get closer to the value of 2 and as we see in the animation As we get closer to the value of X is equal to 2 the function gets closer to the value of 4 now to represent this idea we will use this new mathematical concept called limit and that we are going to represent through the link symbol now We are analyzing What happens to this quadratic function when we get closer to the value of 2 that is, we analyze the limit of the function x squared when x tends to that is, it takes values close to two and this idea is represented by this arrow that will indicate What value x approaches and then indicate what that value is which in this case is 2 also in this first case we start by taking the value of X is equal to 1 which is a value less than 2 and then we went taking values closer and closer to two but less than 2, that is, we are getting closer to two from values that are to the left of 2 and to symbolize this we will place a minus sign as a superscript on the number 2 and this is equal to the value to which the function tends which in this case tends towards the value of 4 in summary we will read it in the following way limit of the function x squared when x tends to 2 from the left is 4 and we will call this type of Limit the left lateral limit and let's analyze the same case again the function x squared but approaching the value of X is equal to 2 from the right let 's suppose that we first take a value of X is equal to 3 replacing the function F of 3 is equal to 3 squared which gives us 9 now we are taking values closer and closer to two and we can see what happens as we get closer to two the function gets closer to the value of 4 therefore using the notation that we saw in the previous example we will have that the limit of the function x squared as x tends to 2 but in this case it approaches two taking values greater than 2 or values that are always to the right of 2 and to represent this we will place a plus sign as a superscript on the number 2 What does it mean that it approaches two from the right and this limit is 4 in summary we would read it as follows limit of the function x squared when x tends to the right is 4 and this other limit we will call the right lateral limit very well and now that we understand the idea behind the limit Let's make an important clarification about the limit since the limit studies What is the tendency of a function When you make the variable tend to a specific value That is, we are not interested in knowing what happens at that value but what happens when you approach that value for example here we have the function x squared again but the function is not defined when x is equal to 2 and this is represented in the Graph with a gap at that point now if we take values close to 2 like making x equal to 3 or equal to 1 and we are getting closer to two we can see that in both cases the function approaches the value of 4, that is to say the two lateral limits both on the left and on the right tend to the same value so we can represent this in the following way the limit of the function x squared when x tends to 2 is 4 and in the previous case both lateral limits coincided but we will not always find ourselves in this situation let's look for example at this case here we have the Graph of this function FX let's see how this function behaves as we approach the value of 2 let's also notice that this function is not defined at x equal to 2 but it does not matter since the limit analyzes what happens when you approach the value of 2 very well let's start approaching from the left, that is to say the limit of this function FX when x tends to 2 from the left we can see how this limit tends towards the value of 1.5 now let's see what happens when we approach two but from the right, that is to say we take the limit of the FX when x tends to 2 from the right and as we can observe this limit tends to 0.5 as we can see in this case the left side limit is not the same as the right side limit so we can ask ourselves if in this case there is a limit of the function with dx tends to 2 So to answer this question let's see In which cases we can affirm that a limit exists and in which cases it does not suppose that we have a function FX whose graph is shown here and we will analyze what happens to this function when we approach the value of a we can also notice that the function is not defined at x is equal to and as we said before it is not necessary for the function to be defined at that value since the limit is an analysis of the tendency towards a specific value in this case in a very general way we will see What is the tendency when x approaches the value of a first let's start by seeing What happens when we approach from the left and as we see in the animation as we approach the value of a the function tends to approach the value of l symbolically the limit of the function F of x when x tends to from the left is l now let's see What happens when we approach from the right and as we see in the animation the function tends to get closer again to the value of l this means that the limit of the function F of x when x tends to the value of a is l and we can see how both lateral limits coincide and when this situation happens we will say that the limit at x that tends to the value of a exists and we will represent it in the following way limit of FX when x tends to the value of a is l in a few words we will say that a limit exists when both lateral limits exist and coincide in the same value, that is, the limit must be unique in a very formal way we can represent it like this the limit of a function FX when x tends to the value of a is l Yes and only Yes the lateral limit on the left is equal to the lateral limit on the right, that is, both lateral limits are equal to the value of l And this is the necessary condition for the existence of a limit if any of the lateral limits were different then we will say that the limit does not exist from here we can conclude that the limit is unique in addition, this other condition is that the function must tend to a finite number when x tends to the value of a, that is, the limit cannot be infinite because infinity is not a number but represents an idea that a you can make it bigger and bigger and bigger and bigger indefinitely but to better understand let's look at the following situation let's see for example how the function FX behaves it is equal to 1 divided by x squared whose graph looks like this this function is not defined when x is equal to zero because when replacing we would get 1 divided by 0 squared which gives us 1 divided by 0 and This division by zero is not defined but we see a curious behavior as x approaches zero for example if we take values of X close to zero from the left we can see how the function gives us increasingly larger and larger values and to represent this we will do it in the following way limit of the function 1 divided by x squared when x tends to zero from the left is infinite this last does not mean that the limit exists and is infinite since as we spoke before for a limit to be seen its result must be a finite number And in this case it does not pass the limit as you approach zero it becomes increasingly larger and larger so this Limit notation in this case is used precisely to convey that idea that as more and more you you get closer to the value of zero from the left the value of the function tends to return larger and larger values and on the other hand if we take values close to zero but from the right we will see how the function returns to giving us larger and larger numbers and we represent this in the following way limit of the function 1 between x squared when x tends to zero to the right is infinity and I repeat again for this case the limit also does not exist but it is customary to use the notation but to help us understand this idea that as you take values closer to zero the function will learn to return larger and larger values and in addition you can draw a dotted line that passes through x is equal to zero here we represent the yellow asymptote and it represents the value that the function can not take in addition you can see how the graphs tend to get closer to this line which is called an asymptote but they can never touch this asymptote and here we have another case of Limit that does not exist in this case we have the function FX is equal to 1 between x minus 12 which is not defined when x is equal to 2 therefore we will not be able to do x equals 2 but we can take values very, very close to two so let's see what happens if we get closer to two from the right we can see how the function tends to return larger and larger values and this idea is represented like this limit of the function 1 between x minus 2 when x tends to 2 from the right is infinity and I repeat this is an abuse of the notation since the limit does not exist and rather we must interpret it this way with the function 1 between x -2 you take values close to two from the right or greater than 2 the function returns larger and larger values and on the other hand if you take values close to two but from the left the function will tend to return increasingly more negative values and this is represented in this way limit of 1 between x minus 2 when x tends to 2 from the left is minus infinity which alludes to the idea that it tends to increasingly more negative numbers also at x equals 2 we can also graph the asymptote of this function as we can see here in the animation and now we're going to remember some previous concepts that will allow us to better understand the formal definition of Limit here we have a number line let's suppose we want to find the distance of this segment that observing we can easily say that it measures 3 units but as we did this calculation then we can do it in the following way since we are on the number line we can use the coordinates of the ends, that is to say the value of 4 and the value of 1 and perform a simple subtraction, that is 4 - 1 which gives us 3 and as we can see this coincides with the distance of the segment but in mathematics to be more strict when taking distances using coordinates this subtraction is done but the absolute value or the module is taken since this operation will guarantee that the result is always positive so that it corresponds to the distance So something more correct would be to say that the distance is the absolute value of 4 - 1 which is the absolute value of 3 which since it is already positive will continue to be 3 but it can also be done in the opposite order, that is to say subtracting 1 - 4 this subtraction will give you minus 3 but a distance cannot be a negative number and for that reason it is necessary to take the absolute value since this will give us the absolute value of -3 which is equal to 3 Always remember that the absolute value will return the positive value and that is why it is useful in this case of measuring distances simply using the coordinates let's look at another example let's suppose we want to find the distance between the point -3 and 4 using the method that we learned this distance would simply be the modulus of three minus four or the subtraction of coordinates operating we obtain the absolute value of -7 and since the absolute value will turn it into positive we will obtain that this is equal to 7 units and the order does not matter For example it could also be the absolute value of 4 - parentheses minus 3 operating on what's inside it becomes 4 + 3 and we are left with the absolute value of 7 which since it is already positive returns the same thing or 7 and as you can see the order does not matter the absolute value of the difference in coordinates will give us the distance between those two points Now I want us to look at this idea in a more general way let's suppose we want to find the distance between these two points whose coordinates are a and x and for this case the distance will simply be equal to the absolute value of X minus a or it could also being the absolute value of A minus x taking into account these concepts we are ready to understand what is behind the famous definition of the limit So let's get started And this is probably one of the most important moments in this channel because we are going to understand what is behind the famous formal definition of Limit and for this let's take a function FX whose graph we can see here in this case when we evaluate the function at point a we obtain the value of l but we are not interested in what happens to the function when x is equal to the value of a but rather what happens to the function when x tends to the value of a we know that if we give values to x that are close to the value of a the function will return values of FX that are close to the value of l And these output values of the function are different from l however we can see this small variation or departure from the value of l as if it were a small error now we are going to establish some condition to limit this error in our approach to the value of L and for this we will establish a quota for this error we could assume that this Bound for example is 0.5 therefore the extreme upper value would be equal a L plus 0 5 and the lower end would be equal to L -0.5 it is important to note that this Bound is always presented in a symmetrical manner in general we can represent this Bound of the error by means of the Greek letter epsilon in this way the upper end will be the plus epsilon and the lower end will be l Minus epsilon now that we have established this quota for the error we can see that the outputs of the function are limited within this interval let's suppose that for a certain value of X we obtain the corresponding FX as we see in the Graph this value of F of X is a small distance from the value of l this small difference would be the error in our approach to the value of L and this error is calculated as the distance that exists between the output value of the function F of x and the value of l that is to say value resulting from FX minus L and on the other hand the Bound of the error that we imposed is equal to epsilon the condition that must be met here is that the error in our approach to the value of l must be less than the Bound of the error or in other words absolute value of F of X minus l is less that epsilon and this error in the approach must be greater than or equal to zero since this absolute value represents a distance between two points so it is always greater than or equal to zero now because we have imposed a quota for the error these limits for the error are directly related to some abscissa values or input values of the function and as we see in the Graph they are not always symmetric there will be some cases in which they are symmetric but for this graph in particular they are not therefore we have two extreme abscissa values that are directly related to the error notes that we have imposed what we will do now is declare an interval that is centered on x is equal to being symmetric and that guarantees that its output values are within our error quota and to guarantee this we will choose the smallest interval And we will call this distance Delta we can see that the distance between the value of X that we used in our approximation and the value of a is equal to the absolute value of X minus a and this distance in turn has to be less than Delta also this distance must be strictly greater than zero because the value of X tends to the value of a or gets closer much towards the value of a but it will never be equal to the value of a now what we are looking for is that the limit of the function FX when x tends to the value of a is l The question is how do we make the value of the function F of X get closer to the value of L and the key to guarantee this is to restrict the value of epsilon or make the value of epsilon smaller each time since by making the value of epsilon smaller the output values of the function are increasingly restricted and therefore they will get closer and closer to the value of l And if we restrict the value of epsilon even more the output values of the function will continue to take values closer and closer to it therefore we have to make the value of epsilon smaller and smaller so that the function gets closer and closer to the value of L therefore to guarantee that the outputs of the function get closer and closer to the value of l this must be fulfilled for every value of epsilon greater than 0 and as small as you want and in turn that this condition is met for every positive value of epsilon it implies that there is a value of Delta also greater than zero therefore to ensure that the limit of the FX function when x tends to the value of a is l these two conditions must be met for every positive value of epsilon as small as desired in short the formal definition of Limit guarantees that you can get close to a specific value and that the function approaches a value of L and now we can declare in a single statement the formal definition of limit as follows given a function FX that is defined in an open interval that contains the value of a taking into account that it is not a necessary condition that the function is defined at point a since we are interested in analyzing what happens when we approach that value then we will say that the limit of the FX function when x tends to the value of a is l if for every positive error bound or for every positive epsilon it happens that the distance between the output values of the function and the value of l is less than the error bound or epsilon when this condition is met there must exist a positive Delta so that the distance between the values of x and the value of a is less than that value of Delta and also in greater steel and in 1905 Albert Einstein published the special theory of relativity without a doubt a theory that would change the way we understood the universe until that moment distances shorten Time passes more slowly energy is transformed into matter and matter into energy bodies acquire more when moving through space and nothing can move faster than the speed of light in a vacuum and here we have the expression for the force in special relativity F is equal to Gamma multiplied by the mass times the acceleration and this factor Gamma appears in many of the equations of relativity and is known as the Lorenz factor which is defined as follows Gamma is equal to 1 divided by the square root of 1 - V squared between C squared where represents the speed of light and v the speed at which a body moves if we graph the Lorens factor we obtain the following for my speed is very low the equations of Newton's dynamics are recovered But as we approach the speed of light the effects of relativity begin to be noticed more and more and as we see this graph has an asymptote when V is equal to c so as the speed tends to the value of the speed of light the Gamma factor tends to infinity and this imposes a limit of the universe that the speed of light is the limit If you had a body that moves and it gets faster and faster as its speed approaches that of light the Gamma factor tends to infinity so that body would need an infinite force to maintain its acceleration making the speed of light an unattainable limit so nothing can move faster than the speed of light and it is exciting to decipher the laws that govern our universe and even more so that we can decipher the greatest mysteries of the universe through mathematics and yes there is no doubt that mathematics has been and is a key and fundamental piece for the development of modern science and technology but there are still many more things to discover and that will give us a greater understanding of how it works our universe and on this path mathematics will always be present like that lantern that lights the way, the path of knowledge. Thank you very much for your attention and see you in the next video [Music]