The following is a simple calculation I was trying to genearlize that I figured was just an application of l'hopitals rule.
Let $r >0$
1) Why is $ \frac{\exp{\left( -\frac{1}{4y^{2r}} \right)}}{y^r} $ bounded for $ y \in [0,1]$?
2) What if I have a different power of $y$ in the denominator $ s>0$ what are the conditions I need on $r,s$ to guarantee $ \frac{\exp{\left( -\frac{1}{4y^{2r}} \right)}}{y^s} $ is still bounded on $[0,1]$?