From Sequence to Function

Calculus Level 4

It is a well known fact that if $\sum a_i$ converges, then $\lim a_i = 0$ .

True or false:

If $f$ is a continuous function such that $\int_0^\infty f(x) \, dx$ exists and is finite, then $\lim_{x \rightarrow \infty } f(x) = 0 .$

True False

1 solution

Calvin Lin Staff
Dec 3, 2016

[This is not a complete solution.]

It is easy to construct an alternating function counterexample, where the alternating parts get smaller and smaller. An example is $\sin x^2$ .

The "true" form of the statement is

If $f$ is a uniformly continuous function such that $\int_0^\infty f(x) \, dx$ exists and is finite, then $\lim_{x \rightarrow \infty } f(x) = 0 .$