Consider the following question:
Prove that $$\frac{x} { \log \log (x) - \log \log \log(x) } + {2}^{ \frac{x} { \log \log (x) } } = O \left(\small\frac{x} { \log \log(x) }\right)$$
It is clear that $\frac{x} { \log \log (x) - \log \log \log(x) }$ is O($\frac{x} { \log \log(x) })$.
I can't prove $ 2^{\frac{x} { \log \log x}} $ less than or equal to any constant as $\frac{x} { \log \log x} $ is neither increasing nor decreasing.
And I don't have any other argument in my mind. So, please help.
It's obviously false: $2^y > C\, y$ for $y$ big enough and $x/\log\log x \to \infty$