Calculus: Understanding Infinity in Functions

[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?

 

[HOI]

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.

 

[highmath]

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?

 

[HOI]

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

 

[highmath]

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?

 

[HOI]

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.