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

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

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

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.

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 0.98507 – 0.2 0.99335 –0.1 0.99833 undefined ÷ by 0 0.1 0.2 0.3 1 We see in the table as x 0 1  

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

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

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

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)

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.

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

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

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

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:

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 0.97355 Undefined 0.1 0.2 0.3 1 = 1

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

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:

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

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) −1 0 2 =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)

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

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

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

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

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

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  

lim 𝑥→0 sin 𝑥 tan 𝑥 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 3. 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

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

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

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

𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 𝟐 = 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐 𝒙 𝒙 𝟐 =𝟏×𝟏=𝟏 6. 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