Evaluar el límite de integración $\lim_{x \to \infty} (\int_0^x e^{t^2} dt)^2/\int_0^x e^{2t^2} dt$

Question

Cómo evaluar $$\lim_{x \to \infty} \dfrac {\bigg(\displaystyle\int_{0}^x e^{t^2} dt \bigg)^2} {\displaystyle\int_{0}^x e^{2t^2} dt} =\ ?$$ ¿Podemos aplicar la regla de L'Hospital?

Answer 1

Claro que sí: $$\lim_{x \to \infty} \dfrac {\bigg(\displaystyle\int_{0}^x e^{t^2} dt \bigg)^2} {\displaystyle\int_{0}^x e^{2t^2} dt} =\lim_{x \to \infty} \dfrac {2e^{x^2}\displaystyle\int_{0}^x e^{t^2} dt } { e^{2x^2} }=\lim_{x \to \infty} \dfrac {2\displaystyle\int_{0}^x e^{t^2} dt } { e^{x^2} }=\lim_{x \to \infty} \dfrac {2\displaystyle e^{x^2} } { 2xe^{x^2} }=0\ .$$