I'm solidifying my calculus by going through Keisler's book that uses a hyperreal/infinitesimal approach. I'm stuck on this problem.
Given infinitesimals $\epsilon,\delta > 0$, deterimine whether the following expression is infinitesimal, finite but not infinitesimal, or infinite:
$$\frac{\epsilon + \delta}{\sqrt{\epsilon^2 + \delta^2}}$$
Keisler gives this hint: Assume $\epsilon \geq \delta$ and divide through by $\epsilon$.
Being an odd number problem, the answer is in the back, but I don't understand how to get it. Also, why would I assume that $\epsilon \geq \delta$?
Well, my take. If $\epsilon \ge \delta$, then $\frac{\epsilon+\delta}{\sqrt{\epsilon^2+\delta^2}} = \frac{1+\frac{\delta}{\epsilon}}{\sqrt{1+\delta^2/\epsilon^2}}$. Now $\delta \le \epsilon,$ so $\delta/\epsilon \le 1$ (because the resulting sequence will have smaller numbers divided by larger numbers a number of times that is in the ultrafilter).
Thus it should be finite but not infinitesimal.