Download notes 1_3 evaluation limits analytically

Notes 1.3: Evaluating Limits Analytically
Goals for today:
1. Evaluate a limit using properties of limits.
2. Develop and use a strategy for finding limits.
3. Evaluate a limit using dividing out and rationalizing techniques.
4. Evaluate a limit using the Squeeze Theorem.
In the last section, you learned that the limit does not depend on the value of f at x = c. It may happen,
however, that the limit is precisely f(c). In that case, the limit can be evaluated by direct substitution.
That is,
.
These types of functions are continuous at c.
Theorem 1.1 Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
1.
2.
3.
= cn
Ex 1: Evaluate the following:
a.
b.
c.
Theorem 1.2 Properties of Limits
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following
limits:
and
1. Scalar multiple:
2. Sum or difference:
3. Product:
4. Quotient:
5. Power:
Ex 2: Evaluate
Theorem 1.3 Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
.
If r is a rational function given by r(x) = p(x)/ q(x) and c is a real number such that q(c)
.
0, then
Ex 3: Find the limit:
Theorem 1.4 The Limit of a Function Involving a Radical
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is
even.
=
Theorem 1.5 The Limit of a Composite Function
If f and g are functions such that
then
Ex 4: Find the limit of each:
a.
b.
Theorem 1.6 Limits of Trigonometric Functions
Let c be a real number in the domain of the given trig function.
1.
2.
3.
4.
5.
6.
Ex 5: Find each limit:
a.
b.
c.
Theorem 1.7 Functions that Agree at All But One Point
Let c be a real number and let f(x) = g(x) for all x
in an open interval containing c. If the limit of g(x)
as x approaches c exists, then the limit of f(x) also exists and
Ex 6: Find the limit:
Ex 7: Find the limit:
The 2 above are called a dividing out technique.
Ex 8: Find the limit:
This is the rationalizing technique.
Theorem 1.8 The Squeeze Theorem
If h(x)
for all x in an open interval containing c, except possibly c itself, and if
then
exists and is equal to L.
Using the above leads to the following theorem.
Theorem 1.9 Two Special Trigonometric Limits
1.
2.
=1
=0
Ex 9: Find the limit:
Ex 10: Find the limit:
Homework: Pgs 65-67 #1-117 odds.