I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I need some help with question 39 in Section 1.5 problems.
$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite?
The expression is this:
$$\frac{\epsilon + \delta}{\sqrt{\epsilon^2 + \delta^2}}$$
It also gives the following hint:
(Hint: Assume $\epsilon \ge \delta$ and divide through by $\epsilon$)
The answer is finite.
I got some semblance of a solution, but I really don't understand fully how to communicate it or how rigorous it is. I think I need some schooling on this one...
$$\frac{1+\frac{\delta}{\epsilon}}{\frac{\sqrt{\epsilon^2 + \delta^2}}{\epsilon}}$$
Because $\epsilon \ge \delta$ we know that the radicand is no greater than $2\epsilon^2$
\begin{align} \frac{1+\frac{\delta}{\epsilon}}{\frac{\sqrt{\epsilon^2 + \delta^2}}{\epsilon}} & = \frac{1+\frac{\delta}{\epsilon}}{\frac{\sqrt{2\epsilon^2}}{\epsilon}} \\ & = \frac{1+\frac{\delta}{\epsilon}}{\frac{\epsilon\sqrt{2}}{\epsilon}} \\ & = \frac{1+\frac{\delta}{\epsilon}}{\sqrt{2}} \end{align}
$\frac{\delta}{\epsilon}$ What is this part? Finite? I don't know if this term is finite or infinitesimal. Either way the result is finite.
The suggested alternative solution doesn't address this question about $\frac{\delta}{\epsilon}$. Also my general curiosity about the logic of my approach is not address in the alternative question.
Please correct me here if I got to the answer the wrong way and fill me in on my missing understanding.
thanks.