Let $n \in \mathbb { N }$. Show that there exists a pair of real numbers $( \alpha _ { n } , \beta _ { n } )$ such that for $x \rightarrow + \infty$,
$$\varphi _ { n } ( x ) = \frac { \alpha _ { n } } { \sqrt { x } } \cos \left( x - \beta _ { n } \right) + O \left( \frac { 1 } { x \sqrt { x } } \right)$$