Chapter 12 Limits
12.1/12.2 Intro to Limits and Evaluating Limits
A. Obj: to evaluate limits, to see if a limit exists, to evaluate
one-sided limits
B. Facts:
1. Limits are the spine that holds the rest of the Calculus
skeleton upright. Basically, a limit is a value that tells
you what height (y-value) a function is headed for or
intended for, as you get close to or approach a specific
x-value.
2. It describes the behavior of the function as it gets
closer to a particular value of x.
3. Formal Definition of a Limit:
a. For a fnc., f(x), and a real number, c, the limit of
f(x) exists if and only if: lim f(x) exists. There must
be a limit from the left.
b. lim f(x) exists. There must be a limit from the right.
c. lim f(x) = lim f(x). The limit from the left must
equal the limit from the right. (Note: this does not
apply to limits as x approaches infinity!)
4. Evaluating Limits:
a. Look for a pattern in a series or a table
b. Simple substitution
c. Use Algebra and/or graph
12.4 Evaluating Limits at Infinity
A. Obj: to evaluate limits at Infinity
B. Facts:
1. Limits @ Infinity for Rational Fncs.
a. If the degree of the numerator and denominator
are the same, then there is a horizontal asymptote
and a limit at the coefficient of the numerator over
the coefficient of the denominator.
b. If the degree of the numerator is larger, then there
is a limit at ∞ .
c. If the degree of the denominator is larger, then
there is a limit at 0.
2.
3.
12.5 Limits
A. OBJ: to evaluate limits of summations
B. Facts:
1.
|r| < 1
2.