$$proof: \forall x > 0, \text{there exists a unique} \quad \xi_x \quad s.t. \int_{0}^{x}e^{t^2}dt=xe^{\xi_x^2} ; \text{solve} \lim_{x \to \infty} \frac{\xi_x}{x} $$
my thought is to use the First mean value theorem for definite integrals to find the$\xi_x$ and assume another $\xi'_x$ to proof the existence and the uniqueness.But for the another problem my thought is $$\lim_{x \to \infty} \ln e^{(\frac{\xi_x}{x})^2 }=\lim_{x \to \infty}\ln(\int _{0}^{x}e^{t^2}dt)^\frac{1}{x^2}$$
but after using L'Hopida's Law, I have no idea what to do next