find a positive continuous function with a finite area : $\int_0^\infty f(x) dx$ , but the $f(x)\rightarrow$ doesn't exist.

Question

find a positive continuous function with a finite area : $\int_0^\infty f(x) dx$ , but the limit of $f(x)$ as $x$ goes to infinity doesn't exist.

I tried finding such a function but I failed .

Welcome to MSE, remember to include your work on the problem, otherwise it looks like you are trying to get others to do your homework. — jjagmath, Apr 18 '21 at 11:14

score 1 · Accepted Answer · answered Apr 18 '21 at 11:10

Hint - try looking at a function who's graph looks like a sequence of triangles who's base gets smaller and smaller, so that the sum of areas of the triangles is finite, but the limit of $f$ does not exist.

If you want your function to be strictly positive, then you can add to it a positive factor which decays to 0 (like a decaying exponential).

find a positive continuous function with a finite area : $\int_0^\infty f(x) dx$ , but the $f(x)\rightarrow$ doesn't exist.

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