A High Limit Horn Tip Problem 140

Give a little personal opinion:

First of all, the triangle column is convergent, which you only need to know by using the two-dimensional case of the closed-interval theorem. The previous triangle must fall completely within the next triangle, which is a true inclusion relationship, and the triangle can be regarded as a closed region on a plane, and an infinite number of such closed regions must be encased in a point, but this point is not a special point, so you have to count it.

The specific calculation is as follows:

Based on the coordinates of the three vertices of the triangle, calculate the inner expression, and then iterate layer by layer. Because the heart of the triangle is always contained in the triangle, you can think of it as the inner convergence of this column to that point. In fact, there are two ways to find this point when you write this iterative formula:

1. Labor-based solution: After a finite iteration, the limit behavior is investigated. Theoretically possible.

2. Intellectual-type solution: The limit point is actually the fixed point of the iterative equation, note that this fixed point may not be unique, but it must contain the point you want. At this time, it is enough to analyze it through the fixed point theory.

Because I forgot the inner formula, and the process is more cumbersome, I don't have the patience to do it, so I only give the solution method, and you can practice the specific operation if you are interested. You might need to use numerical algebra, numerical analysis.

Sure enough, it's a great question, take a seat and watch the master answer.

(0) Contains (1) Contains. Contains (k)Contains (k+1)Contains.

k tends to infinity, and s( (k)) tends to 0;

So (k) converges with the only one point. (Interval Nest Theorem).

Taken apart into the sum of the two limits.

The former makes use of the law of Lopida.

The latter one is deformed using a universal formula.

The Limit -2 process is as follows:

Regardless of whether x tends to 0+ or 0-, the limit is 0-sin0=0, because x and sinx have limits, so you can substitute them directly.

x 0x and sinx are infinitesimal of equal order.

So this limit is 0.

I don't know who the people who write the book now! I am also a freshman, so it is recommended to find classic textbooks or English textbooks from good schools.

It can be done by using the equivalent infinitesimal when x tends to 0, e x and x+1 are equal infinitesimal and e x x+1; x and sinx are equal infinitesimal which accompaniment, x sinx a x=e (xlna) xlna+1 a sinx=e (sinxlna) sinxlna+1 after substitution is equal to limx 0[(xlna+1)-(sinxlna+1)] sinx) 3 =limx 0[lna(x-sinx)] sinx) 3 then using Robida's rule, the derivative of the numerator and denominator is obtained twice to obtain the following result limx 0sinxlna [6sinx(cosx) 2-3(sinx) 3] After about a sinx, Li Shidu's substitution of x=0 can return the silver, and the final result is LNA 6

1. This question is a fixed formula, and you can substitute it directly. Even if the answer after substitution is infinity, it is still a formula;

For the answer to this question, please refer to the first sheet below.

2. For the specific calculation method of the limit, please refer to the summary example below. Due to the huge size of the article, it is not possible to upload all of them. The limit calculation method starting from the second ** is more than enough to cope with the graduate exam.

3. If you have any questions, please feel free to ask, answer any questions, and explain any doubts;

The answer must be meticulous, the interpretation must be exquisite, and the picture must be exquisite, until satisfied.

4. All ** can be fixed, and the enlarged ** will be very clear.

Please be considerate and do not certify. Thank you for your understanding! Thank you for your understanding! Thank you! Thank you!

Directly substituting into the calculation, this is not zero to zero, infinity to infinity.

Hello! Divide the top and bottom by x to find the limit of 1, note the sign before the radical. The Economic Mathematics team will help you solve the problem, please adopt it in time. Thank you!

This kind of question is what I am most afraid of in the exam!

It's not that I'm afraid of how difficult it is in this kind of question, but that the teacher's language description is vague and the level is unclear!

Listening to classes is also the most afraid, hateful, and hated this kind of teacher!

Every sentence is vague, every concept is muddy, and the more you learn, the more tired you are!

Analysis of this question].

1. The meaning of this question is nothing more than to test: monotonous and bounded sequences must have a limit, that is, convergence.

2. Monotony + bounded, combined into one, is a sufficient condition for convergence sufficiency;

Monotony is one of the conditions, a necessity;

Bounded is also one of the conditions, and it is also a necessity.

Monotony and boundedness are both necessary conditions for convergence and sufficient conditions for convergence together.

In this way, monotony and boundedness are the sufficient and necessary conditions for convergence.

sufficient and necessary condition]

3. Among the known conditions of this question, the only thing missing is "bounded", and the simple "bounded" is only a necessary condition, not a sufficient condition; However, when "bounded" is combined with the known conditions of the topic, the whole is a sufficient condition, because all the terms are necessary conditions, and together they constitute the condition of dividing by the sufficient.

Logical problems in the description of this question].

a. This question is a question:

What is the condition of "bounded" alone?

The answer is: a necessary condition, not a sufficient condition!

b. Still asking:

Combined with the condition of "bounded", what is the condition?

The answer is: it is a sufficient condition, and it is also a sufficient condition!

The landlord understands the bastard teacher, where the bastard is?

In addition, it should be noted that this question is incremental, that is, monotonous]:

Because all an is positive, adding one item increases a little, so it is incremental...

Choose a, which is the basic theorem of the convergence of positive series.

Original = lim[-2sin((sinx+x) 2)sin((sinx-x) 2)] x 4....and differential product;

Cover this x 2-(sinx) 2] 2x 4....and so on, the price is infinitesimal and infinitesimal;

Three times Luo will be eliminated.

sin2x）/（12x）

sin2x/ （6*(2x)）