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

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