1. Making it formal (#limitscontinuity, #theoreticaltools) Our goal here is to give a more precise definition for limits and continuity, and use those to prove that a limit exists. The formal definition of a limit of f(x) at point x = a goes as follows: For every positive value e, there exists another value, 8, such that the following is true: if the distance between the x value and our point of interest, a, is less than 8, then the distance between our function at that point f(x) and the limit value, L, is less than e. Let's see how this works in an example. (a) Consider the function f(x) = 3x - 4. Find the value of lim f(x) using the techniques we discussed in class. We will call that value L. (b) Now, we will prove that the lim f(x) = L, where L is the value you found in I-2 € part (a), by verifying that the definition above holds. Say you are given a value of = 1. What value of 6 would guarantee us that if the distance between x and our point of interest, 2, is less than 6, then the distance between f(x) and L is less than 1? We can write this question with equations: for what value of 8 would we have |x2|< 8 imply that |f(x) - L| <1? (c) Draw a diagram that illustrates the scenario you analyzed in part (b). Make sure to clearly indicate where x, 6, and € are in your diagram. (d) Try to generalize this: make a table with various values of € and the correspond- ing value of that would satisfy the conditions above. Use the observations from your table to come up with a formula that relates & and €.