Question

Every magnitude which grows continually but not beyond all limits
must certainly approach a limiting value.”

We can interpret “magnitude” as a real number x that changes in time, thus, as a real valued function x(t), t ∈ R. The above sentence is a theorem about the behaviour of such a function as t approaches ∞.

(a) The assumptions of the theorem are that the function x(t) is monotone (“grows”) and is bounded (“not beyond all limits”). Give a precise definition of each of these properties of the function x(•).

(b) State the theorem in your own words.

Answer 1

@goformit100

Answer 2

@amistre64

Answer 3

i dont know what this question is talking about any help or hints would be really appreciated

Answer 4

It's basically giving you a function where the domain pertains to all real numbers, or in other words in x(t), t cannot be complex.
It is also describing x(t) as limited, or that parameter t cannot or will not exceed a certain value. However, the function is monotone, so it either increases or decreases to some extent.

To define its properties, you simply state a fact about x(t).
Example: "x(t)'s domain is all real numbers."

And to state the theorem in your own words, understand the theorem so that you can use it in a real application.

If you need any further help, I can provide another example of your choosing.

Answer 5

but i dont get what x(.) means it has no formula in my textbook thats what i didnt get. Like I couldnt find the "Precise definition"

Answer 6

x(.) is just a simple way of stating \[x : R \rightarrow R\] It's basically x(t), but it gives the parameter differently.

Answer 7

so the precise definition is just x:R --> R ?

Answer 8

and in my own word its a mapping of real numbers to real numbers?

Answer 9

@Voidus