The discussion centers on the behavior of the mean value of a function as T approaches infinity, specifically analyzing the expression \(\frac{1}{T}\int_{0}^{T}f(x)dx\). Participants conclude that if this limit equals zero, it does not necessarily imply that the function f(x) approaches zero as x approaches infinity, particularly in the case of oscillating functions like \(f(x) = \sin(x)\). The integral of oscillating functions remains bounded while the denominator increases, leading to a limit of zero. This highlights the distinction between the mean value and the pointwise behavior of the function.
PREREQUISITESMathematicians, calculus students, and anyone interested in the behavior of functions and their integrals, particularly in the context of limits and oscillation.
I have no clue what you mean by that. What does ' mean?zetafunction said:let be a function so [tex]\frac{1}{T}\int_{0}^{T}f(x)dx =0[/tex]
as T--->oo does it man that the function f(x) tends to ' as x--->oo
Well, did you try an example? What if f(x)= sin(x)?what if 'f' is an oscillating function ??