#### Infrared

Science Advisor

Gold Member

- 353

- 112

This isn't true, unfortunately. A continuous function with smaller and smaller "spikes" can be integrable on ##[0,\infty)## and yet fail to have a limit at infinity.In order for this integral ##\lim_{t\to\infty}\int_0^t x_{p}(t)\, dt## to converge to a finite value we must have that:

$$\lim_{t\to\infty}x_{p}(t)=0$$

Edit: thanks for noticing that ##p>0## is necessary.

Last edited: