Suppose f(x) is a piecewise continuous function defined for [a,∞).
If an inproper integral ∫[a,∞)f(x)dx coverges, what can we say on a limit(x→∞) f(x)?
Since we assume the inproper intgegral caoverges, we can say there is some c such that
inf(x>c)f(x)c)f(x)>=0
otherwise ∫[c,∞)f(x)dx>=inf(x>c)f(x)×(∞-c)=∞ or ∫[c,∞)f(x)dxc)f(x)×(∞-c)=-∞ that contradicts the assuption.
So we can say
1, if limit(x→∞) f(x) converges,it converges to 0,
2, limit(x→∞) f(x) can't diverge.
What else can we say?
For example, If limit(x→∞) f(x) oscillates around 0 without convergence perpetually, is it possible that the improper integral ∫[a,∞)f(x)dx converges?
And how about on others?
Best regards