If the series {Xn}, {Yn} and {Zn} meet the following conditions:

(1) when n >; When N0, where N0∈N* has Yn≤Xn≤Zn,

(2){Yn} and {Zn} have the same limit a, let-∞ < a <; +∞

Then, the limit of the sequence {Xn} exists when n→+∞, limXn =a a.

It is proved that because limYn=a and limZn=a, according to the definition of sequence limit, for any given positive integer ε, there are positive integers N 1 and N2. When n >; When N 1, there is 〡 yn-a ∣ ∣ ε, when n >; In N2, there are ∣ Zn-a ∣ ε, N=max{N0, N 1, N2}.

So when n> ∣ Ina ∣?

When F(x) and G(x) are continuous at X0 and have the same limit a, that is, x→X0, limF(x)=limG(x)=A,

Then if there is a function f(x)≤f(x)≤G(x) in a certain neighborhood of X0,

Then when x approaches X0, there is limF(x)≤limf(x)≤limG(x).

That is, a ≤ LIMF (x) ≤ A

Therefore, LIMF (x0) = a.

Simply put: function A>b, function b > C, the limit of function A is X, and the limit of function C is X, then the limit of function B must be X, which is the pinch theorem.

2. Monotone bounded criterion: monotonically increasing (decreasing) a series with upper (lower) bounds must converge.

When using the above two to find the function limit, we should pay special attention to the following points. First, we should prove the convergence by monotone bounded theorem, and then find the limit value. 2. The key to applying the pinch theorem is to find functions with the same limit value, and when they meet the limit value, they tend to the same direction, thus proving or finding the limit value of the function.

3. Cauchy convergence criterion

The necessary and sufficient condition for the sequence {Xn} to converge is that there is always a positive integer n for any given positive number ε, so that when m >; N, n> When n and m≠n, there is

We call {Xn} satisfying this condition a Cauchy sequence, so the above theorem can be expressed as: the sequence {Xn} converges if and only if it is a Cauchy sequence.

Extended data

Functional limitations can be divided into

But the definition of ε-δ is more common in the proof of known limit values. Mastering this kind of proof is of great benefit to beginners to deeply understand the definition of application limit.

Take the limit of x→x0 as an example. The definition of f(x) whose limit is at point x0 is that for any given positive number ε (no matter how small it is), there is always a positive number.

So when x satisfies the inequality,

, the corresponding function values f(x) all satisfy the inequality:

Then the constant a is called the limit of the function f(x) when x→x0.

The key to the problem is to find the one that meets the definition requirements, and some inequality techniques, such as scaling, will be used in this process. 1999 directly examines the examinee's mastery of the definition.

Such as the uniqueness of the function limit (if the limit exists, the limit at this point is unique).

Baidu Encyclopedia-Function Limitation

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