How can we prove: $$ \operatorname{erfi}(x) \in O (e^{x^2}) $$
Samely we can prove: $$ \lim_{ x \rightarrow \infty } \frac {\int_0^x e^{t^2} \, dt} {e^{x^2}} = 0 $$
Also it is so awkward. The integral of a function should be greater than or equal to that function in growth rate but here it seems like it is the opposite.
By the answer here, we can observe this, but it's still vague that how that answer is achieved.
It's important to mention that the relations mentioned are all achieved via WolframAlpha.
You say that 'the integral of a function should be greater than or equal to that function but here it seems like it is the opposite'; the idea that an integral is bigger than the function itself is intuitive, but $e^{x^2}$ grows so quickly that it outpaces its accumulated integral up to whatever point of evaluation. The exponential function is (sometimes definitionally!) the balancing point with respect to rate of growth; it grows exactly fast enough to keep up with its integral, up to the requisite constant. Anything that grows faster than $e^x$ will outgrow its integral in the same way that $e^{x^2}$ does.