I have the next problem.
$$f(x)=\frac{e^{-(x-\alpha)/\beta}}{\beta\left ( 1+e^{-(x-\alpha)/\beta} \right )^{2}}$$
and
$$F(x)=\frac{1}{1+e^{-(x-\alpha)/\beta}}$$
I need calculate
$$h_{X}(t)=\lim_{\epsilon \rightarrow 0}\frac{\mathbb{P}\left ( t\leq X< t+\epsilon |X\geq t \right )}{\epsilon }$$
When I develop this expression (I did it with the definition of conditional probability and the inequalities of the probability part) I got: $$\mathbb{P}\left ( t\leq X< t+\epsilon |X\geq t \right )=$$ $$\frac{\left ( \mathbb{P}\left ( X\geq t \right ) \right )\left ( \mathbb{P}\left ( X<t-\epsilon \right )-\mathbb{P}\left ( X<t \right ) \right )}{\mathbb{P}\left ( X\geq t \right ) }=$$ $$ \frac{1}{1+e^{-\left ( t+\epsilon -\alpha \right )/\beta}}-\frac{1}{1+e^{-\left ( t-\alpha \right )/\beta}}$$
And then I just calculate the limit $$h_{X}(t)=\lim_{\epsilon \rightarrow 0}\dfrac{\frac{1}{1+e^{-(t+\epsilon -\alpha)/\beta}}-\frac{1}{1+e^{-(t-\alpha)/\beta}}}{\epsilon }$$
But when I try to calculate the limit, this limit doesn't exist. I don´t know where my mistake is. I hope someone can tell me where my mistake is and why.