Hello fellow CSDN users, today, Xiao Yalan is learning about extreme arithmetic.
review
infinitesimal limit algorithm
Theorem 1: The sum of two infinitesimals is infinitesimal
Theorem 2: The product of a bounded function and an infinitesimal is infinitesimal
Four extreme arithmetic rules
Theorem 3
Theorem 4
Theorem 5: Order preservation in the limit
example
1. Limits of polynomials (organized functions)
2. Limits of rational fraction functions
Limit arithmetic rules for composite functions
Summary of Extreme Arithmetic Rules
Let’s first review the knowledge we have learned before—the limit of a function
It is definitely not enough to rely on the limit of functions to solve problems. There are certain limitations, so we have introduced a series of knowledge points.
We will learn the four arithmetic rules and the limit operation rules of composite functions
From left to right: double limit problem, right limit problem, left limit problem
In the last lesson, we learned about infinity and infinitesimal, and learned about the relationship between infinity and infinitesimal: under the same change process of the independent variable, if f(x) is infinite, then 1/f(x) is infinitesimal, conversely, if f( x) is infinitesimal, and f(x) is not equal to 0, then 1/f(x) is infinite
Our review ends here, let’s get to the point
1. Infinitely small limit algorithm
Theorem 1: The sum of two infinitesimals is infinitesimal
This is the proof process of our Theorem 1. Next, let’s strike while the iron is hot and look at a few examples.
Beginners can easily make this mistake, so what does the correct solution look like?
Theorem 2: The product of a bounded function and an infinitesimal is infinitesimal
Let’s take a look at the proof process
This is the proof process of our Theorem 2. Next, we have several corollaries.
Extending Corollary 2, we can see that the product of a finite number of infinitesimals is also infinitesimal.
Then, let’s look at some examples
sin x is a bounded function, and 1/x is infinitesimal. This example completely explains that the product of a bounded function and infinitesimal is infinitesimal.
Some people will write like this
This way of writing is not rigorous, because when x tends to infinity, the limit of sin x does not exist
Looking at this example, |cos 1/x |<=1, so cos 1/x is a bounded function. This is another problem of the product of a bounded function and infinitesimal
Let’s look at another example
After the above study, everyone already knows that the sum, difference and product of two infinitesimals are all infinitesimal . So some people must be thinking: Is the quotient of two infinitesimals infinitesimal?
The answer is no. I believe you can see that the limit of two infinitesimal quotients is very complicated.
2. Four arithmetic rules of limits
Theorem 3
Then let us prove (2)
This A is a constant function, β is an infinitesimal, the product of the constant function and the infinitesimal is infinitesimal, so the limit of A times β is infinitesimal; similarly, B is a constant function, α is an infinitesimal, then B times α The limit of is also infinitesimal. α is an infinitesimal, β is also an infinitesimal, and the product of two infinitesimals is infinitesimal, so the limit of the product of α times β is infinitesimal.
Here are two points to note about this theorem:
(1) When applying Theorem 3, pay attention to the conditions: lim f(x), limm g(x) both exist
Let’s talk about two corollaries below.
Theorem 4
Theorem 5 (order preservation in the limit)
3. Example questions
1. Limits of polynomials (organized functions)
Let’s look at some simple questions
2. Limits of rational fraction functions
Let’s look at some small examples
in conclusion
general conclusion
a topic
At this time, remember that you must not expand the molecule, as this will require too much calculation! ! ! Directly look at the ratio of cubic term coefficients
4. Limit operation rules of composite functions
example
Summary of Extreme Arithmetic Rules
Okay, that’s it for Xiao Yalan’s learning today, and she will continue to work hard in the future! ! !