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1 INTERESTING CANCELLING

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2 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

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3 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.

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4 Graph Y 1 = xy = -0.30.98507 -0.20.99335 -0.10.99833 0undefined ÷ by 0 0.10.99833 0.20.99335 0.30.98507 1 0 First we will examine the value of for values of x close to 0. We see in the table as x 0 1

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5 Graph Y 1 = Since and then

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6 We need to review some trigonometry before we can proceed to the proof that Slides 7 to 13 are included for those students interested in looking at the formal proof of this limit. We will now move on to slide 14

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7 The Circle x y r x 2 + y 2 = r 2

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8 Unit Circle (0,-1) (-1, 0)(1, 0) (0, 1) x 2 + y 2 = 1 (cos θ, sin θ)

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9 Areas of Sectors in Degrees Area of circle = r 2 If θ = 90 o then the sector is or of the circle.

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10 Areas of Sectors in Radians 360 o = 2 radians

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11 (1, 0)(cos θ, 0) (0, 1) O A BD C cos θ, sin θ (0, sin θ) r = cos θ r = 1 The size of ∆OAB is between the areas of sector OCB and sector OAD

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12 Area of sector OCB Area of ∆OAB Area of sector OAD < < < << < divide by ½ divide by cos << < < <<

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

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14 Example 1: Estimate the limit by graphing -0.30.7767 -0.20.8967 -0.10.97355 0Undefined 0.10.97355 0.20.8967 0.30.7767 x 01 = 1

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15 If the coefficients on the x are equal the limit value will be 1

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16 Example 2: Evaluate the limit Solution: Multiply top and bottom by 2: Separate into 2 limits: Evaluate

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17 Example 3: Evaluate the limit Solution: Multiply top and bottom by 3: Separate into 2 limits: Evaluate

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18 -0.030.015 -0.020.01 -0.010.005 0Undefined 0.01-0.005 0.02-0.01 0.03-0.015 00 THE SECOND IMPORTANT TRIGONOMETRIC LIMIT We see in the table as x→0x→0 →0→0

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19 Mathematical Proof for Multiply top and bottom by the conjugate cos x + 1 Pythagorean Identity sin 2 x + cos 2 x = 1cos 2 x – 1 = – sin 2 x

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20

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21 Example 4: Evaluate the limit

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22 Example 5: Evaluate the limit

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23 Example 6: Evaluate the limit

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24 EXAMPLE 7: REMEMBER: Solution

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25 ASSIGNMENT QUESTIONS 1. 2

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26 2. 9

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27 3.

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28 4. Use your calculator to estimate the value of the following limit. x-0.2-0.1-0.0100.010.10.2 1.6511.9111.999ERR1.9991.9111.651 2 0 2

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29 Algebraic Method

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30 ASSIGNMENT QUESTIONS 5. Multiply by the conjugate Remember cos 2 x + sin 2 x = 1 so cos 2 x – 1 = –sin 2 x Substitute

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31 6.

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