Evaluating the limit $\lim\limits_{x\rightarrow 0}\frac{\int_{0}^{\sin{x^2}}e^{(t^2)}dt}{x^2}$ I am to find the $$\lim_{x\rightarrow 0}\frac{\int_{0}^{\sin{x^2}}e^{(t^2)}dt}{x^2}$$ and I have no idea how to approach this. As its widely known, there's no way to integrate $e^{(x^2)}$, how should I do it? All kind of tips would be appreciated!
 A: The function $t\mapsto e^{t^2}$ is increasing on $(0,\infty)$, hence for all $x$
$$\frac{\sin x^2\cdot 1}{x^2}<\frac1{x^2}\int_0^{\sin x^2}e^{t^2}\,\mathrm dt<\frac{\sin x^2\cdot e^{\sin^2(x^2)}}{x^2} $$
Noting that $\frac {\sin t}t\to 1$ as $t\to 0$, and by contuínuity $e^{\sin^2x^2}\to 1$, we see that the limit in question is $1$.
A: I'm going to take a guess, given the form of the integral and of the limit, that the party who posed the problem to darenn was up to something a bit snarky.
We can define the integral function $$ \ F(x^2) \ = \ \int_{0}^{\sin{x^2}}e^{(t^2)}dt \ \ . $$  
The limit  $ \ \lim_{x \rightarrow 0}  \frac{\int_{0}^{\sin{x^2}}e^{(t^2)} \ dt}{x^2} \ $ can then be written as
$$ \lim_{x \rightarrow 0} \  \frac{( \int_{0}^{\sin{x^2}}e^{(t^2)} \ dt ) \ - \ ( \int_{0}^{\sin{0^2}}e^{(t^2)} \ dt ) }{x^2} \  , $$
which now resembles the limit definition of a point derivative.  Hence, we have
$$ \lim_{x^2 \rightarrow 0} \  \frac{F(0 + x^2) \ - \ F(0)  }{x^2} \ = \ F'(0) \ = \ e^{[\sin(0^2)]^2} \ \cdot  \ \cos(0^2) \ = \ e^0  \cdot \cos(0) \ = \ 1 \ , $$
by applying "Liebniz' Rule". $ ^{*} $
$ ^{*} $ at least, this is what I've sometimes seen the formula for differentiating an integral function, with an integration limit being a function, called... There is something else that seems to be more usually called "Liebniz' Rule".
EDIT: Oops, did that a bit too quick and neglected the Chain Rule; fortunately, that other factor is also 1 . (There is also a bit of abuse here, since I am differentiating with respect to $ \ x^2 \ $ , so the Chain Rule does not produce yet another factor after $ \ \cos(x^2) \ $ ... )
A: I'd recommend using L'Hopital's Rule and The Fundamental Theorem of Calculus
