Compute $\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^{2}} dy]^2}{\int^x_0 e^{2y^{2}}dy}$
You rather have $$\frac{\mathrm{d} }{\mathrm{d} x}\left(\int^x_0 e^{y^2}\mathrm{d}y\right)^2 = 2\left(\int^x_0 e^{y^2}\mathrm{d}y\right) * \left(\frac{\mathrm{d} }{\mathrm{d} x}\int^x_0 e^{y^2}\mathrm{d}y\right)=2\color{red}{\left(\int^x_0 e^{y^2}\mathrm{d}y\right)}(e^{x^2})$$ giving $$ \lim_{x \rightarrow +\infty} \frac{2\color{red}{\left(\int^x_0 e^{y^2}\mathrm{d}y\right)}}{e^{x^2}}=\lim_{x \rightarrow +\infty} \frac{2e^{x^2}}{2xe^{x^2}}=\lim_{x \rightarrow +\infty}\frac1x=0. $$