1. U3 L2 Limits of Primary Trig Functions
UNIT 3 Lesson 2
Limits of Primary Trig Functions

2. U3 L2 Limits of Primary Trig Functions
INTERESTING CANCELLING
1 𝑛 sin 𝑥=?
1 𝑛 sin 𝑥=?
𝑠𝑖𝑥=6

3. LIMITS OF TRIGONOMETRIC FUNCTIONS
U3 L2 Limits of Primary Trig Functions
LIMITS OF TRIGONOMETRIC FUNCTIONS
In order to understand the derivatives that the trigonometric functions will produce, we must first understand how to evaluate two important trigonometric limits.
The first one is
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙

4. U3 L2 Limits of Primary Trig Functions
The Sandwich Theorem
First evaluate something that we know to be smaller
Second evaluate something that we know to be larger.
Make a conclusion about the value of the limit in between these small and large values.

5. U3 L2 Limits of Primary Trig Functions
First we will examine the value of sin 𝑥 𝑥 for values of x close to 0.
Graph Y 1 = sin 𝑥 𝑥 x
Y 1 = sin 𝑥 𝑥
– 0.3
– 0.2
–0.1
undefined ÷ by 0
0.1
0.2
0.3 1
We see in the table as x

6. U3 L2 Limits of Primary Trig Functions
Graph Y 1 = sin 𝑥 𝑥
Since and then

7. U3 L2 Limits of Primary Trig Functions
We need to review some trigonometry before we can proceed to the proof that
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙
Slides 7 to 14 are included for those students interested in
looking at the formal proof of this limit.
We will now move on to slide 15

8. U3 L2 Limits of Primary Trig Functions
The Circle x y r
x 2 + y 2 = r 2

9. U3 L2 Limits of Primary Trig Functions
Unit Circle
x 2 + y 2 = 1
(0,-1)
(-1, 0)
(1, 0)
(0, 1)
(cos θ, sin θ)

10. Areas of Sectors in Degrees
U3 L2 Limits of Primary Trig Functions
Areas of Sectors in Degrees
Area of circle = p r2
If θ = 90o then the sector is or of the circle.

11. U3 L2 Limits of Primary Trig Functions
Areas of Sectors in Radians
360o = 2p radians

12. U3 L2 Limits of Primary Trig Functions
(1, 0)
(cos θ, 0)
(0, 1) O A B D C
cos θ, sin θ
(0, sin θ)
r = 1
r = cos θ
The size of ∆OAB is between the areas
of sector OCB and sector OAD

13. U3 L2 Limits of Primary Trig Functions
Area of sector
OCB
Area of
∆OAB
OAD
<
<
<
divide by ½
<
divide by q cosq
<
<

14. U3 L2 Limits of Primary Trig Functions
In order to evaluate our limit, we now need to look at
what happens as θ→0
REMEMBER: cos 0o = 1
As we approach this limit from the left and
from the right, it approaches the value of 1.
Conclusion:

15. Example 1: Estimate the limit by graphing
U3 L2 Limits of Primary Trig Functions
Example 1: Estimate the limit by graphing x
-0.3
0.7767
-0.2
0.8967
-0.1
Undefined
0.1
0.2
0.3 1 = 1

16. Example 2: Evaluate the limit
U3 L2 Limits of Primary Trig Functions
Example 2: Evaluate the limit
Solution: Multiply top and bottom by 2:
Separate into 2 limits:
Evaluate

17. Example 3: Evaluate the limit
U3 L2 Limits of Primary Trig Functions
Example 3: Evaluate the limit
Solution: Multiply top and bottom by 3:
Evaluate
Separate into 2 limits:

18. U3 L2 Limits of Primary Trig Functions
THE SECOND IMPORTANT TRIGONOMETRIC LIMIT
-0.03
0.015
-0.02
0.01
-0.01
0.005
Undefined
-0.005
0.02
0.03
-0.015
We see in the table as
x→0 →0

19. U3 L2 Limits of Primary Trig Functions
Mathematical Proof for
U3 L2 Limits of Primary Trig Functions
𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝒙−𝟏 𝒙
Multiply top and bottom
by the conjugate cos x + 1
−lim 𝑥→0 sin 𝑥 𝑥 lim 𝑥→0 sin 𝑥 cos 𝑥+1
lim 𝑥→0 cos 𝑥−1 𝑥 × cos 𝑥+1 cos 𝑥+1
−1 sin (0) cos 0 +1
lim 𝑥→0 cos 2 𝑥 −1 𝑥( cos 𝑥+1)
− =0
Pythagorean Identity
sin 2 x + cos 2 x = 1
cos 2 x – 1 = – sin 2 x
𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝒙−𝟏 𝒙 =𝟎
lim 𝑥→0 −sin 2 𝑥 𝑥( cos 𝑥+1)

20. Example 3: Evaluate the limit 𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝟐𝒙−𝟏 𝟐𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝟐𝒙−𝟏 𝟐𝒙
lim 𝑥→0 cos 2𝑥−1 2𝑥
× 𝐜𝐨𝐬 𝟐𝒙+𝟏 𝐜𝐨𝐬 𝟐𝒙+𝟏
𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝟐 𝟐𝒙 −𝟏 𝟐𝒙 (𝐜𝐨𝐬 𝟐𝒙+𝟏)
𝐬𝐢𝐧 𝟐 𝟐𝒙+ 𝐜𝐨𝐬 𝟐 𝟐𝒙 =𝟏
𝐜𝐨𝐬 𝟐 𝟐𝒙 =−𝐬𝐢𝐧 𝟐 𝟐𝒙+𝟏
𝐥𝐢𝐦 𝒙→𝟎 − 𝐬𝐢𝐧 𝟐 𝟐𝒙 𝟐𝒙 (𝐜𝐨𝐬 𝟐𝒙+𝟏)
𝐜𝐨𝐬 𝟐 𝟐𝒙−𝟏 =−𝐬𝐢𝐧 𝟐 𝟐𝒙
𝐥𝐢𝐦 𝒙→𝟎 − 𝐬𝐢𝐧 𝟐𝒙 𝟐𝒙 × 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝐜𝐨𝐬 𝟐𝒙+𝟏
−𝟏× 𝐬𝐢𝐧 𝟐(𝟎) 𝐜𝐨𝐬 𝟐(𝟎)+𝟏
−𝟏×𝟎=𝟎

21. Example 4: Evaluate the limit
U3 L2 Limits of Primary Trig Functions
Example 4: Evaluate the limit
𝐥𝐢𝐦 𝒙→𝟎 𝟐 𝐜𝐨𝐬 𝒙−𝟐 𝟓𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝟐 (𝐜𝐨𝐬 𝒙−𝟏) 𝟓𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝟐 𝟓 × 𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝒙−𝟏 𝒙
𝟐 𝟓 ×𝟎=𝟎

22. U3 L2 Limits of Primary Trig Functions
Example 5: Evaluate the limit
𝐥𝐢𝐦 𝒙→𝟎 𝒙 𝐜𝐨𝐬 𝟐𝒙−𝒙 𝟖𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝒙 (𝐜𝐨𝐬 𝟐𝒙−𝟏) 𝟒(𝟐𝒙)
𝐥𝐢𝐦 𝒙→𝟎 𝒙 𝟒 × 𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝟐𝒙−𝟏 𝟐𝒙
𝟎 𝟒 ×𝟎=𝟎

23. U3 L2 Limits of Primary Trig Functions
Example 6: Evaluate the limit
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝒙 𝒙 𝟐
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝒙 𝒙 𝟐 × 𝟏+ 𝐜𝐨𝐬 𝒙 𝟏+ 𝐜𝐨𝐬 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 𝟐 (𝟏+ 𝐜𝐨𝐬 𝒙)
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐 𝒙 𝒙 𝟐 × 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟏+ 𝐜𝐨𝐬 𝒙
𝟏 𝟐 × 𝟏 𝟏+ 𝐜𝐨𝐬 𝟎 = 𝟏 𝟐

24. U3 L2 Limits of Primary Trig Functions
EXAMPLE 7:
𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝟒𝒙
𝐭𝐚𝐧 𝒙 = 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝟒𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝟒𝒙 𝐜𝐨𝐬 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 × 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟒 𝐜𝐨𝐬 𝒙
𝟏× 𝟏 𝟒( 𝐜𝐨𝐬 𝟎) = 𝟏 𝟒

25. 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝒙 × 𝟐 𝟐 = 𝐥𝐢𝐦 𝒙→𝟎 𝟐× 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟐𝒙 = 𝟐
ASSIGNMENT QUESTIONS 1.
𝐥𝐢𝐦 𝒙→𝟎 𝒔𝒊𝒏 𝟐𝒙 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝒙 × 𝟐 𝟐 = 𝐥𝐢𝐦 𝒙→𝟎 𝟐× 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟐𝒙 = 𝟐 2.
𝒍𝒊𝒎 𝒙→𝟎 𝒔𝒊𝒏 𝟐 𝟑𝒙 𝒙 𝟐
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟑𝒙 𝒙 × 𝟑 𝟑 × 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟑𝒙 𝒙 × 𝟑 𝟑
𝟑𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟑𝒙 𝟑𝒙 ×𝟑 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟑𝒙 𝟑𝒙 =𝟗

26. lim 𝑥→0 sin 𝑥 tan 𝑥 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙3.
lim 𝑥→0 sin 𝑥 tan 𝑥
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 ÷ 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 × 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝟎=𝟏

27. 4. Use your calculator to estimate the value of the following limit.
lim 𝑥→0 sin 6𝑥 sin 3𝑥 x
-0.2
-0.1
-0.01
0.01
0.1
0.2
1.651
1.911
1.999
ERR 2 2

28. Algebraic Method lim 𝑥→0 6𝑥 6𝑥 sin 6𝑥 3𝑥 3𝑥 sin 3𝑥
𝐥𝐢𝐦 𝒙→𝟎 𝟔𝒙 𝟑𝒙 × 𝐬𝐢𝐧 𝟔𝒙 𝟔𝒙 𝐬𝐢𝐧 𝟑𝒙 𝟑𝒙
𝐥𝐢𝐦 𝒙→𝟎 𝟐× 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟔𝒙 𝟔𝒙 ÷ 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟑𝒙 𝟑𝒙 = 𝟐

29. U3 L2 Limits of Primary Trig Functions
ASSIGNMENT QUESTIONS 5.
Multiply by the conjugate
Remember cos2 x + sin2 x = 1 so cos2 x – 1 = –sin2 x Substitute

30. 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 𝟐 = 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐 𝒙 𝒙 𝟐 =𝟏×𝟏=𝟏6.
lim 𝑥→0 1− cos 2 𝑥 𝑥 2
𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 𝟐 = 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐 𝒙 𝒙 𝟐 =𝟏×𝟏=𝟏