Lecture 1. The limit of a function at a point
The lecture is devoted to the limit of a function at a point. We have already discussed the concepts of the neighborhood of a point: let x0 be a point of the numeric line, e>0, the e- neighborhood of the point x0 is an interval with the center at the point x0 and the distance from this point to the ends is equal to e.
Now we will consider the concept of a punctured e-neighborhood. If we remove the center from the e-neighborhood, we get the union of two intervals, and we call such a neighborhood a punctured one, it is obtained without a center, without the point x0.
The transition to the inequalities is as follows: the membership of the e-neighborhood is an inequality module: xÎUe(x0) ó |x – x0| < e, and the punctured neighborhood membership is a conjunction of two conditions, or a dual inequality: xÎUe(x0) ó |x – x0| < e Ù x ¹ x0 ó 0 < |x – x0| < e.
Any time we talk about the limit of a function at a point, we will assume that x0 is the limit point of the domain of definition, even if we don’t mention it. What do we mean by this? The point x0 is called a limit point of the set X if whatever the neighborhood of the point x0 is, there are infinitely many points of the set X in it.
Here is the definition of the limit. Again, you are supposed to learn how to write a definition in the language of the neighborhood, learn this definition and be able to read it. The number A is called the limit of the function f at the point x0, if for any e-neighborhood of point A there exists a σ–neighborhood of the point x0 that for any point x of the domain of definition of the function f, if x is in the punctured σ-neighbourhood of the point x0, the value of f at this point is in the e-neighbourhood of the point A: lim f(x) = A, if x®x0 ó "Ue(A)$Uδ (x0)"xÎDf(xÎ Uδ(x0) Þ f(x)Î Ue(A)).
Let’s try to illustrate this definition because, of course, these words are very difficult for us to understand, what is behind them. So, let’s plot the function f, set it graphically, and try to argue graphically that the limit of the function at the point x0 is the number A. We will solve the problem by definition.
The first point: let’s construct the e-neighborhood of the point A. So, the number A is on the y-axis, and the e-neighborhood of the point is also on the y-axis. Next, by definition, we need to search for the σ-neighborhood of the point x0 – this is the next entry in the definition. How do we find this neighborhood? We refer to the last expression, to the last statement in the definition, and find those points on the x-axis for which f(x) is in the e-neighborhood of the point A. Look, we have marked this set on the Ox axis (see the video).
Afterwards, we find the δ-neighborhood of the point x0. What is this set? This is a neighborhood of the point x0, may be without a center, which is located in the marked set – this is a set symmetric with respect to the point x0. It is easy to see that this is the desired δ-neighborhood of the point x0. Indeed, if we take an arbitrary point x from the marked δ-neighborhood, f(x) will be in the e-neighborhood of the point A.
From this illustration, it is not very clear how the function value differs from the function limit. Let’s look at this figure: if the value of the function at the point x0 is not A, but some other value. All the previous arguments remain true, the limit of the function at the point x0 is still the number A. How can we imagine what the limit of a function is? If we move along the graph of the function towards the point x0, we will find ourselves at the point whose ordinate is equal to A. So, the limit is, in a sense, a movement along the graph of a function.
There is another equivalent definition. If that definition in the language of neighborhood, in the language of inequalities is called the Cauchy definition, this definition is in the language of sequence or according to Heine. Again, you can write it down symbolically, or you can and should be able to read it by just saying the words.
So, the number A is called the limit of the function f at the point x0, if whatever the sequence (xn) made up of points in the domain of the function f other than the point x0 is, if the sequence (xn) converges to the point х0, the sequence of values of the function (f(xn)) must converge to the number A.
Keep in mind that x0 must be the limit point of the definition domain of the function f, only in this case the limit definition works.
When do we use this definition? We use it most often when we want to find out that there is no limit to a function.
Let’s try to prove that the function f(x) = sgn x has no limit at the point 0. We have considered such a function, you specify it as follows: (see the video). What are we going to do? We will consider the sequences converging to the point 0, the sequences xn, they will be located on the x-axis.
So, the first sequence is composed of the numbers 1/n, and the other is -1/n, they both converge to the number 0. But if we find the function values at these points, in the first case, we have values always equal to 1, a stationary sequence, its limit is 1, in the second case, the stationary sequence is composed of -1, its limit is -1.
So, we’ve found two sequences that converge to zero, for which the corresponding sequences of function values converge to different numbers. It means that the limit of functions at point 0 does not exist. Here, we have a new case – there is no limit to the function. But if we consider the movement of the function graph from the left and right to the point 0, we will see that we just get to different points on the y-axis – to the point (0; -1) and to the point (0; 1).
In this case, we encounter a new concept – the concept of one-sided limits. Let’s start with just an illustration, define the function f graphically and consider the limit at the point x0. So, if we move along the graph of the function f towards the point x0 on the left, we will come to the point with the coordinates (x0; A), the number A is the limit of the function f on the left, and if we move to the point x0 on the graph on the right, we will come to the point with the coordinates (x0; B), point B is the limit of the function at the point x0 on the right, although the value of the function at the point x0 = C. Just as in the case of sgn, the limit of the function f at the point x0 does not exist, although there are one-sided limits – the numbers A and B.
Let’s try to define one-sided limits more strictly. If the definition of the limit contains the concept of δ-neighborhood and the δ-neighborhoods are punctured, for one-sided limits, we consider those intervals that make up the punctured δ-neighborhoods, we call them the left semi-neighborhood and right semi-neighborhood of the point х0 of the radius δ-δ- semi-neighborhood.
Besides, we have the concept of a right-hand and left-hand limit point here, just as in the general case.
Let’s look at the definitions, the notations, to be exact, that arise when we consider the concept of a one-sided limit.
So, the limit of the function f is at the point x0 on the right, then x0 is considered the right-hand limit point of the domain of definition. The notations for the function limit on the right: 1) lim f(x), при x®x0+0; 2) lim f(x), при x®x0, x>x0; 3) f(x0+0). Look at the three forms of notation that can be found in different sources.
The same is for the limit of the function on the left: 1) lim f(x), при x®x0-0; 2) lim f(x), при x®x0, x<x0; 3) f(x0-0). We are not going to read out the definition of the function limit at the point x0 on the left and right now, look how these definitions are written (see the video). In contrast to the general definition of limit, everything is the same here except that the right semi-neighborhood for the limit on the right is considered instead of the δ-neighborhood, for the limit on the left, the left semi-neighborhood will be considered. This definition is read by the same rules. This is a task for you, please learn these definitions, learn how to write the definition of the limit of the function at the point x0 on the left, on the right, and learn to read these definitions.
Just as for the sequence, the theorem about the uniqueness of the limit of a function at a point is valid: if a function has a finite limit at x0, i.e. a number, this limit is unique. The proofs are very similar, and you can already try to conduct this proof on your own.
How is the concept of a function limit related to one-sided limits? Let x0 be a right-hand and left-hand limit point of the domain of definition, so that we can consider the concept of one-sided limits, a function f at x0 has a finite limit if and only if its two one-sided limits exist, they are finite and equal to the same value.
The following theorem is a necessary condition for the existence of a finite limit of a function at a point. The necessary condition is that the function is bounded in some neighborhood of the point where the limit is considered. This theorem says: if a function at some point has a finite limit, it is bounded in some neighborhood of that point. The arithmetic operations theorem is used to calculate the limits. If the limit of the function f and the limit of the function g at the point x0 exist and are finite, equal to the numbers A and B, at this point, there are limits of the functions f+g, f-g, f∙g, f/g, if B¹0, which are calculated using the specified formulas (see the video).
It is an important theorem that is used in the proof of other theorems, it has such a folklore name as “The two policemen theorem”, and the strict name is “The limit of intermediate functions theorem.” Probably, now I should say some words about the policemen, but this name is old, it originated in the Soviet era, and, in general, it should be clear.
So, let the functions f, h, and g satisfy the double inequality f(x) ≤ h(x) ≤ g(x) in some neighborhood of the point x0, this inequality may be violated at the point x0. If the functions f and g at the point x0 have the limit number A, the function h, which is located between f and g, we see in the inequality, also has the number A as its limit at the point x0.
The limit transition theorem allows us to sometimes avoid errors in our reasoning. So, what is given to us? The limit of the function f at the point x0 is the number A, the limit of the function g at point x0 is B. In addition, it is known that f(x) ≤ g(x) for all x in some neighborhood of the point x0. I must say that ≤ here is a special case when f is strictly < g, the conclusion still remains true, that is, the values of the limits – the numbers a and b, are connected by the inequality ≤. A logical error that is sometimes allowed from the strict inequality f < g is: sometimes you might think that A < B. No! It may turn out that a strict inequality can also turn into an equality of limits, and this is what is said in the above remark.