The discussion revolves around the concept of limits in calculus, specifically addressing the continuity of the function f(x) at a point a. The participants clarify the definitions of ε (epsilon) and δ (delta) in the context of limits, emphasizing that for a limit to exist, the function must approach a specific value L as x approaches a. The example function f(x) is analyzed, demonstrating that the limit as x approaches 1 is 1, not 2, due to the behavior of the function on either side of 1. The importance of one-sided limits and the distinction between the value of the function at a point and the limit itself are also highlighted.
PREREQUISITESStudents of calculus, mathematics educators, and anyone seeking to deepen their understanding of limits and continuity in functions.
The distance form f(x) to L, f(x)- L (not f(x) alone), can be larger than \epsilon for some x, just not for x "sufficiently close" to a. Once x is within some distance of a, |f(x)- L| cannot exceed \epsilon. And that "distance" is, of course, \delta.
Close but not exactly. The value f(x) "reaches" when x reaches a is f(a). The question is what happens when x is close to a but not equal to a.