Function with specific conditions I have to find at least one function $f(x)$ that satisfies the following conditions (or prove that it does not exist):


*

*$$f(x)<1\quad\text{for} \quad x\in \mathcal{R^+}$$

*$$\lim_{x\to +\infty}\quad \frac{1}{1-f(x)}=0$$

*$$\lim_{x\to +\infty}\quad f(x)=0$$
I always end with satisfying only 2 conditions over three.
Any suggestion on how to start?

You can prove that such a function does not exist. $$\lim_{x\to\infty} f(x) = 0 =>\lim_{x\to\infty} \frac{1}{1-f(x)}=1 $$.
