I'm reading Henle/Kleinenberg's Infinitesimal Calculus. Here:
They speak about guessing a $\delta$. Is there an algorithm to find a suitable $\delta$?
2026-03-31 10:05:26.1774951526
Is there an algorithm to "guess" a suitable $\delta$?
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As @ZacharySelk comments, "not really". The author is telling a joke - perhaps one only mathematicians would appreciate. You work the algebra out on scratch paper, figure out what $\delta$ you need to get the $\epsilon$ inequality you want, then write the proof starting with that miraculously useful guess.
This was a particularly easy example. With practice you get better at guessing in hard cases.