Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: dt \rightarrow 0$ as $x \rightarrow 0^+$? Is the reverse implication true?

I suspect that the answer is "no" in both cases, so here's my real (although vague) question: is there a notion of regularity for $f$ (along the lines of the notion of almost-periodicity) such that the two limit-assertions imply each other when $f$ is regular?