So I tutor this student, and we came to the conclusion that the delta needs to be 0.1975, but her TA thinks it should be 0.2025. This was a homework question for her, and she got it wrong because she originally had what I had. Where did I go wrong, and how can you prove it using the epsilon delta proof? Everything I have found, and every attempt at a solution I give says that we were correct…
I found this answer by finding b and c, then finding the distances from x=4. Knowing the nature of $\sqrt{x}$, the shortest distance would need to be delta to avoid being outside the range of epsilon by picking the larger distance. This comes out to the distance from b to 4, which is 0.1975.
I proved this using the standard epsilon delta proof and it worked out. If needed, I can go through my steps here
Best Answer
The TA's answer is clearly incorrect, because if $\delta = 0.2025$, you can choose $x = 3.8$, which satisfies the condition $|x - 4| < \delta$, but not $|\sqrt{x} - 2| < 0.05$. If the TA cannot accept this simple counterexample, then they are not competent to grade this question.