proofs about limits

Proof about limit is a hard type of questions we prove. Briefly, limf(x) (x->a) = l is equivalent to "for all e > 0, there exists a d > 0, for all x in real numbers, if |x - a|<d then |f(x) -  l|<e". Because it states that for all e > 0 in the implication, we need to write d in form of e while we pick the d, and that is the most difficult part in the proof.

Sometimes it is very easy to choose d for a certain type of function ( such as f(x) = x). However, sometimes we'll have trouble when picking appropriate d. Danny gave an example about proving limx^2(x->3) = 9, and I don't want to repeat the whole process here. In the class, Danny used a very clever way to simplify the process of proving. When rounded |x^2 - 3^2| all the way down to < d(d+6), Danny made full use of the precondition that we can actually pick d whatever we like to. At this step he supposed all the d he pick is less than 1, so he left the left d and turn the make d(d+6) < 7d (#d<1), so to make 7d less than e we just need to pick d: min(e/7, 1). In this way, we won't have to solve the d(d+6) = e to get root and pick d and this largely simplified the process.

I really benefited from this lecture because it solved my question about the problem in maths courses again!(What exactly is the min(.......) about??) After the class I asked Danny how he came up with this method and he told me that it required a little bit of intuition. He said that intuition is required to raise an assumption, to construct the basis of an idea or to find an easier way to the solution, and then the logic follows up to strictly and objectively test the idea and the method, to make sure the intuition leads to the right direction. Actually, we need to combine both intuition and logic to thrive in the domain of science. I learned a lot from his words.