Question

Suppose f is continuous on [0, infinity) and limit of f(x) as x approaches infinity is 1. Is it possible that the integral of f(x) from 0 to infinity is convergent?

Answer 1

yes, yes it is.

Answer 2

\[\lim_{a \rightarrow \infty} \int\limits_{0}^{a} f(x)dx\]

Answer 3

can you show me the steps to proof that it is possible the integral is convergent? cos right now i only have the answer that it is not convergent by using integration by parts

Answer 4

Depending that F(x) integrates into a simple equation in terms of x, then yes it would converge because you would take x, replace it with a, and then use your limit and take it to infinity. Easy.
It would diverge if the infinity would be in the denominator of a quotient and the denominator was going to infinity faster than the numerator. Just look at the big picture, or "in a long run".

Answer 5

If 1/x^p and p > 1, then it CONVERGES.

Answer 6

hi quantish, if you take f(x) as 1/x^p and p>1, what does it converge to? I get a divergent answer when i integrate that from 0 to infinity. Only when I integrate it from 1 to infinity do i get a convergent answer. thanks so much for all the help everyone.

Answer 7

I was just refering to the p-series test, and wasn't refering to a function in particular and only a limit as x --> infinity. If the denominator increases at a greater rate than the numerator i.e.
1/x^p when p > 1, then it should CONVERGE, not DIVERGE as was earlier stated.