Limits and Continuity
Limit existence vs. Continuity of a Function
This is a dynamically interactive graphical application intended to highlight the definitions of Continuity and Limit Existence.
Graphical examples and non-examples of the epsilon-delta definition of a limit as x approaches 1.
One can illustrate failures of the definition when the limit exists but does not equal the candidate value being tested and when the limit does not exist. Comparing the three examples side-by-side illustrates how the definition of continuity encapsulates the intuitive graphical description.
You may select and deselect the checkmarks located under the graphs to observe the delta-neighborhoods around x=1 and the epsilon-neighborhoods around suggested values of A, B, C for f(x), g(x), h(x), accordingly.
Continuity of f at p
Definition 1: A function f is continuous at p if and only if f is defined at p and the limit of f(x) as x approaches p exists and equals the number f(p).
Question 1: Which of these functions is continuous at 1? Why or why not?
Limit of f(x) as x approaches p
Definition 2: The limit of f(x) as x approaches p exists and equals a number L if and only if for each epsilon-neighborhood of L there exists a delta-neighborhood of p such that the image of the delta-neighborhood of p under f is contained in the epsilon-neighborhood of L.
Question 2: Which of these limits exist and what are their values? Why or why not?