I had posted another answer but my answer didn't really answered your issue. I believe this one will fit much better. The answer frequently is: They proved it before and they remember the suitable $\delta$. To see this is true, just take some very weird and non-obvious limit to your professor and see if (s)he finds a $\delta$ fast.
From my experience teaching mathematics, I guess there is a reason people do this: People try to deduce what is happening in others minds, for example: You've deduced that "your professor did it so fast" not that "he could know the answer", not that "he could have done that thousands of times before."
I've tried to teach mathematics a lot of times pretending the student and me both don't know the answer, I proceed heuristically: "We have this, but what do we need to reach that?" And I try to use tools available to the student, which frequently is nothing. A lot of things in mathematics can be answered almost mechanically with enoughly powerful tools. But without those tools, things can be painful and confusing even for very basic problems, I've seen professors I consider extremely intelligent struggle to solve problems with very basic tools. Do you know what they (these students) ended up thinking about me? That I didn't know how to answer or that I was stupid.
It is much better to be seen as "someone who find $\delta$'s very fast" than as "someone stupid who doesn't know the answer" specially when you do know the answer and the student barely knows how to read the question. Sometimes, other people are more interested in the game of "labeling stupidity" than in learning something.
 : Which can be useful for a student who wants to justify his/her shortcomings.