Say within e , which is some positive number. The way we usually phrase it is to say that someone else , your opponent in the game, chooses e , and you then have to choose a good enough d to win. That just means that x a . Now. let's make it precise. It would be some number, E .
Solution Let E be given.
Now, let's find the number D . The idea is to find a D expressed in terms of E that works, so that, no matter what E is, D is adjusted automatically. Here the way to do it is to square both sided of the inequality. Note that I said that that choice of D will do . Note that this says two things, that the limit exists, and that the function's value at a is the same as the limit.