Section 1.3 - Finding Limits Analytically
Objective - find limits of functions using direct substitution
1
One method to solve limits is through
Direct Substitution
With this method, the functions
must be CONTINUOUS.
2
Theorem 1.1 - Some Basic Limits
(Where b and c are real numbers and n is a positive integer)
1.
2.
3.
See Thm 1.2 on page 57!!!
3
Theorem 1.5 - The limit of a composite function
If f and g are functions such that and
then
ex:
4
Theorem 1.6 - Limits of Trigonometric Functions
5
Rationalization Technique
ex: Find the limit:
Check direct substitution first!
Then rationalize the numerator...multiply by the conjugate.
6
Cancellation Technique
ex: Find the limit
lim
x ­3
lim
x ­3
2
x + x ­ 6
x + 3
2
x + x ­ 6
x + 3
2
lim (x + x ­ 6) = 0
x ­3
Direct Substitution Fails!!
lim (x + 3) = 0
x ­3
2
lim (x + 3)(x ­ 2) = lim (x ­ 2) = ­5
lim x + x ­ 6
=
x ­3
x ­3 x + 3
x ­3
(x + 3)
( Using Thm 1.7 and Direct Substitution )
7
Homework
pg 65 2-66 eoe
AND we will be doing an exploration before you go!
8