Question

Proving Limits (lim x->2 x^2 =4)

Answer 1

Use the \[\epsilon - \delta \] definition of limit to prove the above limit.

Answer 2

This is an example in my book, but I'm having trouble understanding the reasoning behind their solution.

Answer 3

Solution: You must show that for each E>0, there exists a d (I'm using a 'd' instead of delta simply because it's easier to type) >0 such that |x^2-4| <E whenever 0<|x-2|<d . To find an appropriate d, begin by writing |x^2 -4| = |x^2-4|=|x-2||x+2|

Answer 4

Now, I understand that. It's this part where I'm confused: For all x in the interval (1,3), x+2<5 and thus |x+2|<5. So, letting d be the minimum of E/5 and 1, it follows that, whenever 0..... (And it goes on, but before I type that, I want to understand this.) I would like to know if the interval 1,3 is something that is randomly picked and doesn't matter, or if I should know where that came from. And I'm also unsure of where the "5"s are coming from. Help, please?

Answer 5

I think (1,3) is picked solely because the endpoints are 1 unit away from x=2. It would be the same as choosing (0,4) because the distance from endpoint to center to endpoint is the same. You can pick any two endpoints as long as they're equidistant from x=2; (1,3) happens to a very simple one to work with.