lim (x goes to 0) ( (sinx) / x ) ^ (1/ (x^2) ) = ?
$\displaystyle \lim_{x\to 0}\left(\frac{sin(x)}{x}\right)^{\frac{1}{x^{2}}}$
If we let $\displaystyle t=sin(x)$, we get:
$\displaystyle \lim_{t\to 0}\left(\frac{t}{arcsin(t))}\right)^{\frac{1}{arcs in^{2}(t)}}$
$\displaystyle =e^{\left(\lim_{t\to 0}\frac{ln(\frac{t}{arcsin(t)})}{arcsin^{2}(t)}\ri ght)}$
Using L'Hopital and hammering at it, we get it whittled down to:
$\displaystyle e^{\left(\frac{-1}{6}\lim_{t\to 0}\sqrt{1-t^{2}}\right)}$
$\displaystyle =e^{\frac{-1}{6}}$
You have to apply L'Hopital several times.
Eventually, we get:
$\displaystyle e^{\left(\frac{1}{2}\cdot\frac{\lim_{t\to 0}\sqrt{1-t^{2}}}{\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}$
Now, you can see it?.
I hope you can see that. I don't know how to make it bigger.
This can be achieved using \displaystyle :
$\displaystyle \exp{\displaystyle \left(\frac{1}{2}\cdot\frac{\lim_{t\to 0}\sqrt{1-t^{2}}}{\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}$
or
$\displaystyle \exp{ \left(\frac{1}{2}\cdot\frac{\displaystyle\lim_{t\t o 0}\sqrt{1-t^{2}}}{\displaystyle\lim_{t\to 0} -3\sqrt{1-t^{2}}+\lim_{t\to 0} tsin^{-1}(t)}\right)}$