Does $f(x)$ have limit at infinity?

Let $ f:[1,+\infty)\to[0,+\infty)$ be a function such that $ \int_{1}^{+\infty}f(x)dx$ is convergent. I have 3 questions as follows:

question(1) Does $ f(x)$ have limit at infinity?

question(2) If the answer of question (1) is yes, Is $ \lim_{x\to\infty}f(x)=0$ true?

question(3) If the answers of questions (1) and (2) are NEGATIVE, What can we say about their answers, if $ f$ is continuous function?

How do we, in general, know, verify, or otherwise prove (in meta or in FOL) an alleged infinity exists?

Many outstanding difficult mathematical problems (e.g., abc conjecture, Twin Prime conjecture, “There are infinitely many counter examples of GoldBach Conjecture”, …) would involve some set in which its being infinite or otherwise has been only alleged – not proved. It seems natural then to ask if there’s a generic way to prove (in meta or in FOL) an alleged infinity exists. (Thus the question).

Will the energy of a randomly driver harmonic oscillator grow to infinity or oscillate about a finite value?

The equation of motion for an undamped harmonic oscillator, with driver $ f=f(t)$ is given by: $ $ \ddot{x}+x=f.$ $ Let the initial conditions be given by: $ $ x(0)=\dot{x}(0)=0.$ $ If $ f=\cos(t)$ then the solution is: $ $ x(t)=\frac{1}{2}t\sin(t).$ $ Hence, a resonance is setup and the energy of the oscillator will grow forever. If $ f=\cos(\omega t)$ where $ \omega\ne1$ , the solution is: $ $ x(t)=\frac{2}{\omega^2-1}\sin\left(\frac{\omega-1}{2}t\right)\sin\left(\frac{\omega+1}{2}t\right),$ $ hence, the energy oscillates about some finite value. My question is, if $ f$ were replaced with some continuous random driver where the frequency profile resmbled that of say gaussian white noise, would the energy of the oscillator grow forever or would is oscillate about some finite value?

Does anyone know of a simple function I could replace $ f$ with to generate a continuous white noise driver?