Limits and Continuity
- Prof. Caine
Limit of f(x) as x approaches p
Definition: 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: Which of these limits exist and what are their values? Why or why not?
Continuity of f at p
Definition: 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: Which of these functions is continuous at 1? Why or why not?