calculus - Indetermination in limit of integral $lim_{x rightarrow 0}(int_0^{sin x}e^{xt^{2}}dt big/ int _0^{tan x}e^{-xt^{2}}dt)$

I would like to evaluate the following limit
$$\lim_{x \rightarrow 0}\frac{\int \limits_0^{\sin x}e^{xt^{2}}dt}{\int \limits_0^{\tan x}e^{-xt^{2}}dt}$$

In order to use L'Hospital's rule I obviously need derivatives with respect to $x$. With this including x as a parameter in the upper integration limits, I can reduce the limit to
$$\lim_{x \rightarrow 0}\frac{e^{x \sin^{2} x}\cos x+\int \limits_0^{\sin x}t^{2}e^{xt^{2}}dt}{e^{-x\tan^{2}x}\cos^{-2} x+\int \limits_0^{\tan x}(-t^{2})e^{-xt^{2}}dt}$$

I'm not sure if I'm approaching this in the correct way; so my questions are

1. Am I using L'Hospital's rule correctly in the above? If so



2. How should I proceed from now on? I feel a little stuck.

Any help is greatly appreciated.

Answer

Yes the derivatives of the numerator and denominator are correct (e.g. from https://en.wikipedia.org/wiki/Leibniz_integral_rule). And because the integrals in the new quotient go to $0$ as $x\rightarrow 0$, you can simply plug-in $x=0\;$ and get the limit $\frac{1}{1}$