Conquering the Infinity Limit: Integrals and Exponential Functions

[powerof]

$lim_{x\rightarrow + ∞} \frac{\int^{x^3}_{0} e^{t^2}dt}{x \int^{x^2}_{0} e^{t^2}dt}$​

Attempt at a solution: I don't really know where to start. Any hints?

 

[Curious3141]

L' Hopital's Rule. Product Rule for the denominator. Fundamental Theorem of Calculus to differentiate the integrals. Don't forget that the bounds are functions of $x$, so apply Chain Rule (Leibniz's Rule).

After the first step, you'll end up with an expression that I'll call $L$. Find and simplify $L^{-1}$ with a further application of all those rules. Then deduce what $L$ should be at the limit.

 

[powerof]

Thanks. When I get home I'll get to work.