- How do the values of a function f(x) behave when x approaches a
number c, whether or not f(c) is defined?
- Definition of a Limit
- Informally:
The limit of f as x approaches c is L
if and only if
|f(x)-L| can be made arbitrarily small
as
|x-c| is made sufficiently small. - Formally:
- Graphically:
- Informally:
- Theorem 1, p. 52
- Example: #14, p. 58
- Examples:
- One-sided limits: example, #33, p. 58
- #40, 42, p. 58
- Questions regarding Homework/directed reading handout? Section 2.2 (html) (pdf)
- Theorem 1: consider the limit as x approaches c:
sums, differences, product, and quotients behave exactly as hoped:
The limit of the NOSE of two functions is the NOSE of the two limits.
(replace the word NOSE in the statement above with any of the other vocab words).
- Notes:
- The Theorem requires that the limits exist separately for the two functions at x=c.
- We have to be a little careful with the quotient, just to make sure that the limit of the denominator function is not zero....)
- The "constant multiple law" is deduced from the product law and with Theorem 1 from 2.2: that the limit of a constant function is the constant itself.
- The theorem also applies to one-sided limits.
- Examples:
- #10, p. 62
- #18, p. 62
- #24, p. 63
- Questions over the Directed reading handout, Section 2.3 (html) (pdf)?