Extreme ε-δ Proof of definition Can anyone be specific

Extreme ε-δ Proof of definition Can anyone be specific

Definition of function limit: let function f (x) be defined in a de centered neighborhood at x0. If there is constant a, for any ε> 0, there is always a positive number δ, Make it right |x-xo|

Find a proof of the definition of limit? How to prove that the necessary and sufficient condition of LIM n →∞ xn = a is for any ε＞ 0, interval (a)- ε, a+ ε） There are only a limited number of XN

In fact, it is the definition of limit: there is n > 0. When n > N, there is always | xn-a|

The limit existence criterion is used to prove the positive square root of sequence: the positive square root of 2, (the positive square root of 2 + 2) and the positive square root of [2 + (the positive square root of 2 + 2)]

According to the meaning of the question, if this sequence is an, an > 0, then A1 = root sign 2, a (n + 1) = under root sign (2 + an), that is [a (n + 1)] ^ 2 = 2 + an
Easy to get A2 > A1
[a(n+1)]^2-(an)^2=[a(n+1)+an]*[a(n+1)-an]=an-a(n-1)
According to mathematical induction, a (n + 1) > An is an increasing sequence
It is proved that an < 2
A1 = root sign 2 < 2
Suppose AK < 2, a (K + 1) = root sign (2 + AK) < root sign (2 + 2) = 2
It can be obtained by mathematical induction, an < 2
In conclusion, the existence limit of an is obtained from the limit existence criterion
The solution shows that the limit of an is 2

How to use the mean value theorem of definite integral to find the limit

I just passed a paper from others in Baidu Library. You can download it and have a look. It's very detailed
The search title is the limit in the integral mean value theorem (Yang Yonghong level 05). Doc

Use definite integral to find the limit (just use definite integral, no other method) When n →∞, find [1 / (1 + 1 / n ²）+ 1/（1+2/n ²）+ 1/（1+3/n ²）+...+ 1/（1+n/n ²）】/ n. Don't ask me if I have the wrong number. That's what the book says!

When n →∞, 1 / n ² →0,1/（1+1/n ²） → 1 similarly, the following items tend to 1, so 1 / (1 + 1 / n ²）+ 1/（1+2/n ²）+ 1/（1+3/n ²）+...+ 1/（1+n/n ²） → n, so [1 / (1 + 1 / n ²）+ 1/（1+2/n ²）+ 1/（1+3/n&...

Starting from a vertex of an n-sided shape, we can lead () diagonals into () triangles, and the number of all diagonals is ()

Starting from a vertex of an n-polygon, we can lead (n-2) diagonals into (n-2) triangles. The number of all diagonals is ((n-1) (n-2) / 2)