EVALUATING LIMITS ANALYTICALLY WITH TRIG Section 1.3A Calculus AB AP/Dual, Revised ©2013 [email protected] 6/15/22 06:54 PM 1.3 - Properties of Limits with Trig 1
Page 54
Dec 31, 2015
Page 54. Page 67. Evaluating Trig Limits with Trig Functions. Section 1.3. “0/0” Limits AKA: Indeterminate Form. Always begin with direct substitution Completely factor the problem - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 1
EVALUATING LIMITS ANALYTICALLY WITH TRIG
Section 1.3ACalculus AB AP/Dual, Revised ©2013
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 2
“0/0” LIMITSAKA: INDETERMINATE FORM
A. Always begin with direct substitutionB. Completely factor the problemC. Simplify and/or Cancel by identifying a function that agrees with for
all x except = . Take the limit of D. Apply algebra rules
1. If necessary, Rationalize the numerator2. Plug in of the function to get the limit
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 3
EXAMPLE 1
Solve
2
4
16lim
4x
x
x
2
4
4 16lim
4 4x
0
0
What form is this?
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 4
EXAMPLE 1
Solve
8
4
4 4lim
4x
x x
x
4
lim 4x
x
4 4
AS X APPROACHES 4, f(x) OR Y APPROACHES 8.
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 5
EXAMPLE 1 (CALCULATOR)
Solve
8
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 6
EXAMPLE 2
Solve
5
4
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 7
YOUR TURN
Solve
1
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 8
WHEN IN ALGEBRA…
You learned to:
NO RADICALS IN THE DENOMINATOR
IN LIMITS, NO RADICALS IN THE NUMERATOR and DENOMINATOR
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 9
EXAMPLE 3
Solve
9
3lim
9x
x
x
9
9 3lim
9 9x
0
0
What form is this?
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 10
EXAMPLE 3
Solve
9
3 3lim
9 3x
x x
x x
9
3 3 9lim
9 3x
x x x
x x
NO NEED TO FOIL THE BOTTOM
9
9lim
9 3x
x
x x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 11
EXAMPLE 3
Solve
9
9lim
9 3x
x
x x
9
1lim
9 3x
1
3 3
1
6
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 12
EXAMPLE 4
Solve
1
2
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 13
YOUR TURN
Solve Hint: Don’t combine like terms to the denominator, too early
1
4
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 14
EXAMPLE 5
Solve
0
1 15 5lim
x
xx
0
1 15 0 5lim
0x
0
0
What form is this?
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 15
EXAMPLE 5
Solve
0
1 1 lim
5 5 x x
0
1 55lim
5 5 5 5x
x
x x
5
5
5 x 5 x
0
5 5lim
5 5x
x
x
0lim
5 5x
x
x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 16
EXAMPLE 5
Solve
0
5 5limx
xx
x
0lim
5 5 1x
x x
x
0
1lim
5 5x
x
x x
0
1lim
5 5x x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 17
EXAMPLE 5
Solve
0
1lim
5 5x x
1
5 5 0
1
25
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 18
EXAMPLE 6
Evaluate
1
4
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 19
YOUR TURN
Solve
1
16
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 20
“SQUEEZE THEOREM”
A. Also known as the “Sandwich theorem,” it is used to evaluate the limit of a function that can't be computed at a given point.
B. For a given interval containing point c, where , , and are three functions that are differentiable and over the interval where is the upper bound and is the lower bound.
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 21
“SQUEEZE THEOREM”
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 22
EXAMPLE 7
Use the Squeeze Theorem to evaluate where c = 1 for
33 2x g x x
3
3
1 1lim3 lim 2x x
x g x x
3
1 1lim3 1 lim 1 2x x
g x
3 3g x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 23
EXAMPLE 7
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 24
EXAMPLE 8
Use the Squeeze Theorem to evaluate for for which
24 9 4 7x f x x x
7
2
4 4lim 4 9 lim 4 7x x
x f x x x
2
4 4lim 4 4 9 lim 4 4 4 7x x
f x
7 7f x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 25
YOUR TURN
Use the Squeeze Theorem to evaluate where c = 0 for
9
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 26
SPECIAL TRIGONOMETRIC LIMITS
A. B. C. D. When expressing in radians and not in degrees E. The use help explains the “Squeeze” Theorem
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 27
WHY IS THE LIMIT OF (SIN X)/X, WHEN X APPROACHES 0 EQUAL TO 1?
0
sinlim 1x
x
x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 28
WHY IS THE LIMIT OF (1 – COS X)/X, WHEN X APPROACHES 0 EQUAL TO 0?
0
1 coslim 0x
x
x
MEMORIZE IT!
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 29
EXAMPLE 9
Solve
0
tanlimx
x
x
0
sinlim
cosx
x
x x
Is there another way of rewriting tan (x)?
Split the fraction up so we can isolate and utilize a trigonometric limit
0
sin 1lim
cosx
x
x x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 30
EXAMPLE 9
0
sin 1lim
cosx
x
x x
0 0
sin 1lim lim
cosx x
x
x x
Utilize the Product Property of Limits
1
1cos 0
1
Solve
1
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 31
EXAMPLE 10
Solve
0
sin 4limx
x
x
0
sin 4 4lim
4x
x
x
Try to convert it to one of its trig limits.
0
sin 44 lim
4x
x
x
4 4 1
Try to get it where the sine trig function to cancel. Whatever is applied to the bottom, must be applied to the top.
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 32
EXAMPLE 10
Solve
4
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 33
EXAMPLE 11
Solve
2
3
1 sin 2
3
x
x
0
1 sin 2 2lim
3 2x
x
x
0
2 sin 2lim
3 2x
x
x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 34
YOUR TURN
Solve
5
3
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 35
PATTERN?
Solve = 4Solve = Solve = Solve = Solve =
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 36
EXAMPLE 12
Solve
2
0
1 coslimx
x
x
0
1 cos 1 coslim
1x
x x
x
Split the fraction up so we can isolate and utilize a trigonometric limit
0 0
1 coslim lim 1 cosx x
xx
x
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 37
EXAMPLE 13
Solve
0 0
1 coslim lim 1 cosx x
xx
x
0 1 cos 0
0 2
cos(0) = 1
0
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 38
EXAMPLE 14
Solve
20
sin sin coslimx
x x x
x
0
sin 1 coslimx
x x
x x
1 0
0
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 39
YOUR TURN
Solve
0
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 40
AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR)
Solve
(A) π(B) 1(C) 0(D) –1
D
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 41
AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR)
Solve
(A) 0(B) –π/2(C) (2√2)/π(D) 2/π
D
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 42
AP FREE RESPONSE PRACTICE QUESTION (NON-CALCULATOR)
If a ≠ 0, then determine . If the limit does not exist, explain why.
2
1
2a
04/19/2023 08:28 PM 1.3 - Properties of Limits with Trig 43
ASSIGNMENT
Page 6727-36 all, 63-71 all, 73, 89
Related Documents
