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And that is how my textbook taught me to use the epsilon-delta definition of limit to prove the limit of a linear function.
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You can memorize the method that your textbook teaches you to solve a given problem.
There's however the different ability of being able to find your own way to solve a problem. That way requires understanding.
In the end computers can do the work of taking a predefined way to solve a problem a lot better than humans. The ability that computers lack is that they aren't able to come up with their own way of solving new problems.
Part of the ability of being a good mathematican is to develop proofs on your own when you just have definition of axioms and theorems.
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About memorization. A lot of people are echoing the sentiment, "rote memorization is bad". I believe this is a rare example of mathematical dogma.
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The dogma lies in the fact that you don't use a mathematical formula without proof.
Either your professor did the proof during the course do the prove yourself in the exam.
It not just about finding the right answer but also about proving that the answer is right.