# Calculus: Understanding Infinity in Functions

• MHB
• highmath
In summary, if you have a function that is increasing at a given point in space, then you can use calculus to find the limit of the function as x goes to infinity.
highmath
When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?

highmath said:
When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?
So the function is "increasing" or "decreasing". But I have no idea what "in infinity" means. In Calculus, "infinity" is not a number- it makes no sense to talk about the value of a function, or any property of a function "in infinity" or "at infinity". We can talk about the limit of a function "as x goes to infinity".

The most we can say here is that, if a function is increasing, then its limit as x goes to infinity is larger than or equal to any value of the function. If the function is decreasing then its limit as x goes to infinity is less than or equal to any value of the function.

If you are thinking that the limit, as x goes to infinity, of an increasing function must be infinity, that is incorrect. For example, if f(x)= (x- 1)/x= 1- 1/x then f(x) is increasing and the limit as x goes to infinity is 1.

Country Boy said:
function "as x goes to infinity".
.
If I know that x goes to infinity, so how can I know how the function pattern is there?
What the limit help me for?

First, you will have to tell us what you mean by "function pattern".

I know when you draw a function, the value of f(x) is real number always (generalization of natural (N), rational (Q), integer (Z) etc) on the Cartesian System.
So the question is the number theory.
o. k. I will continue with it.

(1)
What axioms I need to prove it?
By what can I use to show that the function is depend on Number Theory?
If I err tell me.
(2)
Is There a calculus way to prove it?
by what means in general?

I think you are confused about basic definitions. "Number theory" deals with specific properties of the positive integers. It is NOT about numbers in general and certainly not the set of all real numbers. In Calculus, a differentiable function is increasing at a given point if and only if its derivative is positive there and decreasing if and only if its derivative is negative.

## 1. What is calculus?

Calculus is a branch of mathematics that deals with the study of change and motion. It is used to analyze and model continuous change in various systems, such as in physics, engineering, economics, and other fields.

## 2. What is infinity in calculus?

In calculus, infinity refers to the concept of a limit, where a function or sequence approaches a value that is infinitely large or infinitely small. It is used to describe the behavior of a function as its input approaches a certain value.

## 3. How is infinity used in functions?

Infinity is used in functions to describe the behavior of the function at certain points or as the input approaches a certain value. It can also be used to determine the end behavior of a function, whether it approaches a finite value or goes to infinity.

## 4. What is the difference between limits and infinity in calculus?

Limits refer to the value that a function or sequence approaches as its input approaches a certain value, while infinity refers to the behavior of the function at that point or as the input approaches a certain value. Limits can approach either a finite value or infinity, while infinity is always a concept of an infinitely large or small value.

## 5. How does calculus help us understand infinity in functions?

Calculus allows us to understand the behavior of functions as they approach infinity, whether it is in terms of growth, decay, or oscillation. It also helps us to determine the limits of functions and their end behavior, which can be crucial in solving real-world problems and making predictions.

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