Download 1.3 - Finding Limits Analytically

AP CALCULUS AB
Section Number:
LECTURE NOTES
Topics: Finding Limits Analytically
MR. RECORD
Day: 4
1.3 (Day 1)
Basic Limits
Let b and c be real numbers and n be a positive integer.
1. lim b  b
2. lim x  c
x c
x c
3. lim x n  c n
x c
Properties of Limits
Let b and c be real numbers and n be a positive integer. Also let f and g be functions such that
lim f ( x)  L and lim g ( x)  K .
x c
x c
1. Scalar Multiple
lim bf ( x)  bL
2. Sum/Difference
lim  f ( x)  g ( x)  L  K
3. Product
lim  f ( x)  g ( x)  L  K
4. Quotient
lim
5. Power
lim  f ( x)  Ln
x c
x c
x c
x c
f ( x) L

g ( x) K
n
x c
Evaluate each of the following limits analytically.
Example 1: Find lim 5 .
x 3
Example 3:
x2  x  4
Find lim
.
x 1
x 1
Example 2:
Find lim 4 x 2  3
Example 4:
Find lim sin x

x 2
x
2
Functions That Agree At All But One Point
Let c be a real number and let f ( x)  g ( x) .
lim f ( x)  lim g ( x)
x c
xc
Example 5:
x2  x  6
x 3
x3
Find lim
x 1  1
x
Example 7:
Find lim
Example 9:
4( x  x) 2  3( x  x)  2  4 x 2  3x  2
Find lim
x 0
x
x 0
x
x 1
Example 10:
Find lim
Example 11:
3x 2  1, x  2
Find lim f ( x) for f ( x)  
x 2
5 x  2, x  2
x 1
x3 1
x 1
Example 6:
Find lim
Example 8:
1
1

Find lim x  3 3
x 0
x
x 1
AP CALCULUS AB
Section Number:
1.3 (Day 2)
LECTURE NOTES
Topics: Finding Limits Analytically
- Limits at Infinity
MR. RECORD
Day: 5
We start this topic with a very important and rather straight-forward theorem.
Limits at Infinity Theorem
If r is a positive rational number and c is any real number, then
c
c
lim r  0 and when possible, lim r  0
x  x
x  x
Example 12: Evaluating a Limit at Infinity
2 

Find the limit lim  5  2 
x 
x 

Example 13: Evaluating a Limit at Infinity
2x 1
Find the limit lim
x  x  1
Example 14: A Comparison of Three Rational Functions
Find each limit.
2x  5
x  3 x 2  1
a. lim
2x2  5
x  3 x 2  1
b. lim
2 x3  5
x  3 x 2  1
c. lim
NOTE: Would it have made any difference in either example above if x approached  ?
**** If you are interested in a shortcut, try looking at the grey box on page 201 of your text. ****
Example 15: A Function Where The Results Differ
Find each limit analytically. Then sketch the function on your TI calculator.
3x  2
3x  2
a. lim
b. lim
x 
x 
2x2  1
2x2  1
Trigonometric Limits – Analytic Approach
Before we discuss trigonometric limits, we first must familiarize ourselves with “The Squeeze Theorem”
The Squeeze Theorem
If h( x)  f ( x)  g ( x) for all x in an open interval containing c, except
possibly at c itself, and if
lim h( x)  L  lim g ( x)
xc
xc
then lim f ( x ) exists and is equal to L.
xc
Special Trigonometric Limits
sin x
1
x
1  cos x
0
2. lim
x 0
x
1. lim
x 0
x
1
sin x
cos x  1
lim
0
x 0
x
lim
or
x 0
Proof of Special Trig Limit #1 Above
Example 16:
Find lim
tan x
x
Example 17:
Find lim
sin 4 x
x
Example 18:
Find lim
x 0
x 0
1  cos 2 x
x 0
x