How to find the limit of a integral?

Question

I have to find the limit of the following integral:

Let $F:\mathbb{R}\rightarrow \mathbb{R}$ and $f$ be a function defined by the following:

$$\int_0^xf(t)\ dt=F(x)\text{ and }f(x)=e^{x^2}$$

Prove that $F(x)\rightarrow\infty$ where $x\rightarrow\infty$

I dont know if I have done it right, but this is what i have reached:

$$\lim_{x\rightarrow\infty}\int_0^xf(t)\ dt=\lim_{x\rightarrow\infty}F(x)=F(\infty)=\infty$$

I hope i can get some help, thanks in advance.

Answer 1

Since, for each $x\in[0,\infty)$, $f(x)\geqslant1$,$$F(x)\geqslant\int_0^x1\,\mathrm dx=x$$and therefore $\lim_{x\to\infty}F(x)=\infty$.

Answer 2

Take $g(t)=e^t$, and then apply comparison test.

$f(t)>g(t)$ as t getting large

You can refer to improper integral section in your calculus textbook.