What values of $\delta$ make this statement true?

Question

Let $\epsilon >0$

Let $\alpha, \beta \in \mathbb{R}$ such that $\alpha > 0$ and

\begin{align*} |1-\alpha| &\leq \delta \\ |\beta| &\leq \delta \end{align*}

For what $\delta > 0$ will \begin{align*} \big{|} \frac{\beta}{\alpha}\big{|} < \epsilon \end{align*} hold?

Answer 1

Since $\alpha\geq 1-\delta$, and $|\beta|\leq \delta$, you get the desired inequality if $$ \frac{\delta}{1-\delta} < \epsilon $$ which happens if $\delta < \frac{\epsilon}{1+\epsilon}$.