Let $f(t)$ be Riemann-integrable and $\int\_{-\infty}^{\infty} f(t) dt=1$.

If I now look at the limit $\lim\limits\_{x \rightarrow \infty}{F(x)}$, where

$$F(x)=\int\_{x}^{x^2} f(t) dt$$ 

I wonder if this limit always has to be $0$ or if it is possible that $\lim\limits\_{x \rightarrow \infty}{F(x)} \neq 0$ ?

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In my opinion this is not possible, since $f(t)$ has to be bounded by definition, thus it has to converge for $x \rightarrow \pm\infty$.

However I'd like to know if the fact, that my Integral goes from $\infty$ to "$\infty^2$" could make some trouble.