 # On the definition of function limit when the independent variable tends to infinity It is defined as "when X - > ∞, the function value f (x) is infinitely close to a certain constant a, then a is called the limit of function f (x) when X - > ∞"   Here, infinite approach means that in the process of X - > ∞, after (at least) a certain point x on the number axis, will the function value be closer and closer to a It means that at least after an absolute value, the greater the absolute value of X, the closer the function value is to a   So why does the following formula hold?   I can understand from the front to the back, but from the back to the front, I don't think it's necessary. The latter two formulas can only ensure the tendency of function values in one direction   For example, suppose the limit value is 6, When x = 3, it is assumed that the function value is   five When x = - 4, it is assumed that the function value is 4   And when | x | - >   When ∞ (from 3 - > - 4, absolute value from 3 - > 4), the function value is not closer to the limit value 6 What is the definition of function limit when the independent variable tends to infinity Can you think of it this way: Let me take some individual points as examples, When x = 1,2,3,4, the function value is 3,4,5,6   (close to limit 10) When x = - 1, - 2, - 3, - 4, the function value is 2, 4,6,8 (or 2,3,4,6) is also close to the limit value 10 Although not strictly according to the absolute value of | x | the greater the absolute value, the closer it is to the limit value 10 In this way, the function f (x) can still be counted as - > a  ?  lim f(x) = A   -------------------- Is it possible to "zigzag forward" towards the limit value of 10

## On the definition of function limit when the independent variable tends to infinity It is defined as "when X - > ∞, the function value f (x) is infinitely close to a certain constant a, then a is called the limit of function f (x) when X - > ∞"   Here, infinite approach means that in the process of X - > ∞, after (at least) a certain point x on the number axis, will the function value be closer and closer to a It means that at least after an absolute value, the greater the absolute value of X, the closer the function value is to a   So why does the following formula hold?   I can understand from the front to the back, but from the back to the front, I don't think it's necessary. The latter two formulas can only ensure the tendency of function values in one direction   For example, suppose the limit value is 6, When x = 3, it is assumed that the function value is   five When x = - 4, it is assumed that the function value is 4   And when | x | - >   When ∞ (from 3 - > - 4, absolute value from 3 - > 4), the function value is not closer to the limit value 6 What is the definition of function limit when the independent variable tends to infinity Can you think of it this way: Let me take some individual points as examples, When x = 1,2,3,4, the function value is 3,4,5,6   (close to limit 10) When x = - 1, - 2, - 3, - 4, the function value is 2, 4,6,8 (or 2,3,4,6) is also close to the limit value 10 Although not strictly according to the absolute value of | x | the greater the absolute value, the closer it is to the limit value 10 In this way, the function f (x) can still be counted as - > a  ?  lim f(x) = A   -------------------- Is it possible to "zigzag forward" towards the limit value of 10

I don't understand why you have to mix the process that x tends to positive infinity with the process that x tends to negative infinity
Why do you think the derivation can't be pushed out on the right and on the left? X on the left tends to infinity, which is not a process, but two processes, that is, the process of tending to positive infinity and negative infinity. The formula on the left actually means that x tends to positive infinity and the limit of obeying infinity is a, which is consistent with the conclusion on the right

### Necessary and sufficient conditions for the existence of function limits when independent variables tend to infinity

For ∀ ε> 0, ∃ a, G > 0, when x > G, | f (x) - A|

### The relationship between higher mathematics function and limit. The book says that the sequence {xn} can be regarded as a function with a positive independent variable n. that is, xn = f (n). Then I Advanced mathematics The relationship between function and limit The book says that the sequence {xn} can be regarded as a function with a positive independent variable n, that is, xn = f (n) Then I have a question. Can function f (n) be regarded as the general term of sequence {xn}? If so, don't all sequences have general terms?

The sequence is not continuous, so it can be regarded as a part of a continuous function, and the function can not be regarded as the general term of the sequence. And subsets are the same truth

### Must there be no limit for unbounded sequence? I am a freshman of huakeda. Recently, I studied calculus and saw a conclusion in the book that unbounded sequence must diverge. I think so. Unbounded sequence may have upper bound, and then I think of a sequence, from negative infinity up to a given positive number. Isn't this the limit of the sequence? Isn't this a convergent sequence? Tangle, tangle, math Xiaobai, seek the truth! Unbounded sequences may have upper bounds, and bounded sequences must have upper and lower bounds

I made a mistake because for the sequence, the size of the first term is certain. No matter how small it is, it is always a certain number rather than an infinite value, which can not be confused with the function. Therefore, no matter how small the first term at the beginning of the sequence is assumed to be n, you can find an n-1 smaller than him as the lower bound. For the problem, because the convergent sequence must be bounded as the true proposition, Therefore, the unbounded sequence of inverse no proposition must diverge into true proposition

### "If the sequence or function has a limit, then the sequence or function must be bounded." in the textbook, I'm stupid and can't understand it. Please ask the teachers for help For example, the function f (x) = x, x = (1, + ∞). The function has a limit at x → 1, but it has no boundary? , hehe This is the definition of boundedness in my textbook, "Let y = f (x) be defined in an interval. If M > 0, there will always be | f (x) for any X in the interval|

Bounded means that on the interval [a, b], when X - > b or a, the function has a limit, which means that the function is bounded. F (x) = x - > + infinite, not convergent
Nor can we deduce that f (x) is bounded

### According to the definition of sequence limit, it is proved that (3n-1) / (2n + 1) [the limit where n tends to infinity] = 3 / 2

The meaning of limit is that infinity tends to a value without saying it is an equal sign. If you say that the limit of this number is 3 / 2, it is absolutely not wrong. If you say that this number is 3 / 2, it is indeed a deviation or even a mistake according to your statement. The definition of limit must be clear