$\underset{x→∞}{\lim}\frac{\left(\int\limits_{0}^{x}e^{x^2}dx\right)^2}{\left(\int\limits_{0}^{x}e^{2x^2}dx\right)}=$

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$\frac{\underset{x→∞}{\lim}\left(\int\limits_{0}^{x}e^{x^2}dx\right)^2}{\int\limits_{0}^{x}e^{2x^2}dx}$   $\left(\frac{∞}{∞}form\right)$

$\underset{x→∞}{\lim}\frac{2\int\limits_{0}^{x}e^{x^2}dx.(e^{x^2}.1)}{e^{2x^2}.1}=\underset{x→∞}{\lim}\frac{2\int\limits_{0}^{x}e^{x^2}dx}{e^{2x^2}}=\underset{x→∞}{\lim}\frac{2e^{x^2}.1}{e^{2x^2}.2x}=\underset{x→∞}{\lim}\frac{1}{x}=0$