Limits of Functions

[K-MOOC] Introductory Mathematics for Artificial Intelligence

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Translated by

Sang-Gu LEE with Youngju NIELSEN, Yoonmee HAM

from the original Korean text written by

Sang-Gu LEE with Jae Hwa LEE, Yoonmee HAM, Kyung-Eun PARK

Part Ⅲ. AI and Optimal solution(Calculus)

7. Limits of Functions

An optimal solution is a solution in which a function defined in a set has a maximum or minimum value. The problem of finding an optimal solution

involves generalized concepts and operations of derivatives. Techniques used to find an approximate solution help when we find an optimal solution.

* History and Concept of Calculus (Storytelling)

http://matrix.skku.ac.kr/Calculus-Story/index.htm

7.1 Limits of Functions

7.2 Derivative and differentiation

7.1. Limits of Functions

Let’s consider the following question. If approaches 1, then what will happen to the value of ?

has a zero denominator when . Hence this function cannot be defined at . We can define the function only if ,

Once we do factorization and cancel the factors of numerator and denominator, it becomes the same as the equation of .

But it only works except for one point at . Therefore, The graph of should be only seen as approaches 1 but not equal to 1.

Now we plot and compute the limit of when approaches to 0 using a simple code. It is much simpler than

drawing a graph by hand. We can use the same code to draw various functions. The exact values are given as follows.

We can find the values of at points near 1, and plot a graph.

In both (left and right) directions of , converges to 2. By using , we can intuitively check and predict approaches 2

when approaches 1. Besides, if 's code 'limit' is used, the limit of the above example is computed as follows.

We can see that when the value of approaches 1 (but not 1), the function converges to 2. We denote this by as

or simply .

묶음 개체입니다.

If converges to as approaches , then we say this "As approaches , converges to " and denoted by .

This is called the limit of . We say diverges if the limit does not exist.

We can change the function and find its limit at any point. In the example below, the graph shows that diverges at .

☞ Note The definition of the limit described above is intuitive. When we plot a graph, we will see where it converges. A precise definition of limit requires

the Epsilon-Delta argument. For more information, please refer to Chapter 2 in the Calculus book. http://matrix.skku.ac.kr/Cal-Book1/Ch2/

Let's compute the right-hand limit and left-hand limit of the function with as described earlier.

Find the limit of the following functions.

(limit),

(left-hand limit), (right-hand limit)

Solution.

(1) = . (2) = -4.

(3) = 4.

Check these result using the following Sage codes.

If both the right-hand limit and the left-hand limit exist, and if the values are the same, we say that there exists a limit of at .

In the previous example, the limit does not exist because the two values are different.

Find, (1) and (2) .

Solution. (1) Find and compare the left and right-hand limits to find .

It is easy to understand when we draw the graph.

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If , then . So, .

The graph of is given as follows.

(2)

(1) The first graph has an infinite value at π/2, so there is no limit.

(2) The second graph is an example of convergence because the limit exists.

☞ Note When we find the limit of the function at a point, it does not matter whether the function has the value at that point.

Even for the case that does not exist, may be defined. If is defined and , then we say that is continuous at .

7.2. Derivative and Differentiation

Let's start with the following question.

Q) Find the equation of the tangent line to at the point .

If the slope of the tangent line is given by at the point of the function , then the tangent line can be obtained by the following formula.

Choose a point near the point on the parabola . Then compute the slope of the secant line . is the slope at (1,1),

In the figure, as the point moves to the point , it is easy to see that the secant line is getting closer to the tangent line.

Therefore, the slope of the tangent line is defined as the limit of the slope of the secant line as . Now we define the derivative with this concept.

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So,=. Hence the equation of the tangent line to at is , which means .

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For a point in the domain of the function , if the following limit

exists, then the function is said to be differentiable at . This limit value is called the derivative(differential coefficient) of function at

and is denoted by .

Consider the tangent line mentioned above. The derivative indicates the slope of the tangent line.

Therefore, the equation of the tangent line to at can be expressed using the derivative instead of , as follows:

If is differentiable at , then must be continuous at . But the converse is not true. In the picture, is not continuous at .

It means is not differentiable at .

묶음 개체입니다.

The above figure shows the interval that the function is continuous and differentiable, the point (point b) that the function is continuous

but not differentiable at the point, and the point (point c) that the function is not differentiable. The slope is infinite at the point.

When is differentiable at every point in an interval, is called differentiable in that interval. In this case, the derivative at that point

is called the derivative of f(x) at . It can be denoted by

Finding the derivative of a function is called the "differentiation of ." The derivative of a function can be found as = .

Therefore, the function obtained by differentiating a function is called the derivative .

If the function of is also differentiable, the derivative can have another derivative .

It is called the second derivative of . and is denoted by . If the second derivative is also differentiable, then the third derivative can be found,

and so on. We denote for the -th derivative of .

If the -th derivative exists, is said to be times differentiable. Speed and acceleration is a good example of the application of the derivatives.

The speed is the first derivative, and the acceleration is the second derivative of the distance function. A constant function is still differentiable,

so a polynomial function is another example of an infinitely-differentiable function.

◆ Fundamental properties of a derivative.

Suppose and are differentiable, then