# Limits and Continuity

- Author:
- Prof. Caine

- Topic:
- Continuity, Limits

## 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?