§1.3: Properties of Limits with Trigonometry Did You… Subscribe to Remind? Text @kphscalbc to 81010 Bring the necessary supplies? If so, would you please leave it on the desk and I will collect it. For today, pick up some notes in the back of the room Fill out the information page (“All About Me”) yet? If not, please complete it by Monday. It is available online at dangmath.com 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Evaluating Limits Analytically Section 1.3 Calculus BC AP/Dual, Revised ©2017 viet.dang@humbleisd.net 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Easy Limits Start with direct substitution Simplify 𝐥𝐢𝐦 𝒙→𝒄 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕=𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝐥𝐢𝐦 𝒙→𝒄 𝒙=𝒄 𝐥𝐢𝐦 𝒙→𝒄 𝒙 𝒏 = 𝒄 𝒏 REMEMBER: IT IS WHAT 𝒙 APPROACHES NOT WHAT 𝒙 IS 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 1 Evaluate 𝐥𝐢𝐦 𝒙→𝟑 𝟐𝒙+𝟓 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 2 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 3 Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙−𝟐 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Review of Compositions Determine what is substituted Take the INSIDE function and replace it Take the outside function and bring it down Replace the variable with the leftover variable Simplify the expression Notation: They may write it as 𝒇 𝒈 𝒙 or(𝒇 ○ 𝒈) 𝒙 . The meaning is the same. 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Review Example A If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒇 𝒈 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Review Example B If given 𝒇 𝒙 =𝟒𝒙 and 𝒈 𝒙 =𝟐−𝒙, solve 𝒈 𝒇 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 4 Find the following limit given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 , determine 𝐥𝐢𝐦 𝒙→−𝟓 𝒈 𝒇 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Find the following limit given 𝒇 𝒙 =𝒙+𝟕 and  𝒈 𝒙 = 𝒙 𝟐 , determine 𝐥𝐢𝐦 𝒙→−𝟓 𝒇 𝒈 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Let 𝒃 and 𝒄 be real numbers, let 𝒏 be a positive integer, and let 𝒇 and 𝒈 be functions with the given limits: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 =𝑳 and 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 =𝑲 Scalar Multiple: 𝐥𝐢𝐦 𝒙→𝒄 𝒃𝒇 𝒙 = 𝒃 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 Sum or Difference: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ± 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 Product: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ∙ 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 Quotient: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 5 Find the following limit given 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) = 𝟕 𝟔 and 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) = 𝟓 𝟔 determine 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) + 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧𝟐(𝒙) 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 7 Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝒙 𝒆 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 8 Given the graph of 𝒇 𝒙 , find 𝐥𝐢𝐦 𝒙→𝟏 (𝟓𝒇 𝒙 + 𝒇 𝟏 ) 𝐥𝐢𝐦 𝒙→𝟒 (𝒇 𝒙 ) 𝟐 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Solve the following limit given 𝐥𝐢𝐦 𝒙→−𝟏 𝒇 𝒙 =𝟑 𝒙 𝟐 −𝟐𝒙−𝟏 and 𝐥𝐢𝐦 𝒙→−𝟏 𝒈 𝒙 = 𝒙 𝟐 +𝟏 , determine 𝐥𝐢𝐦 𝒙→−𝟏 𝒉 𝒙 = 𝒇 𝒙 𝒈 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

“𝟎/𝟎” Limits AKA: Indeterminate Form Always begin with direct substitution Completely factor the problem Simplify and/or Cancel by identifying a function 𝒈 that agrees with for all x except 𝒙 = 𝒄. Take the limit of 𝒈 Apply algebra rules If necessary, Rationalize the numerator or denominator Plug in 𝒙 of the function to get the limit 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 9 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 What form is this? 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 9 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 AS 𝒙 APPROACHES 4, 𝒇(𝒙) OR 𝒚 APPROACHES 8. 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 10 Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐭𝐚𝐧 𝒙 𝐜𝐨𝐬 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

When in Algebra… NO RADICALS IN THE DENOMINATOR You learned to: NO RADICALS IN THE DENOMINATOR FOR LIMITS, NO RADICALS IN THE NUMERATOR OR DENOMINATOR 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 11 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 What form is this? 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 11 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 NO NEED TO FOIL THE BOTTOM 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 11 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 12 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 What form is this? 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 12 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 12 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 12 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 13 Solve 𝐥𝐢𝐦 𝒙→𝒂 𝒙−𝒂 𝒙 𝟑 − 𝒂 𝟑 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry “Squeeze Theorem” 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. For a given interval containing point 𝒄, where 𝒇,  𝒈, and 𝒉 are three functions that are differentiable and 𝒈 𝒙 <𝒇 𝒙 <𝒉 𝒙 over the interval where 𝒇 𝒙 is the upper bound and 𝒉 𝒙 is the lower bound. 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry “Squeeze Theorem” 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 14 Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) where 𝒄=𝟏 for 𝟑𝒙≤𝒈 𝒙 ≤ 𝒙 𝟑 +𝟐 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 14 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 15 Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝟒 𝒇(𝒙) for 𝟒𝒙−𝟗≤𝒇 𝒙 ≤ 𝒙 𝟐 −𝟒𝒙+𝟕 for which 𝒙≥𝟎 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) where 𝒄=𝟎 for 𝟗− 𝒙 𝟐 ≤𝒈 𝒙 ≤𝟗+ 𝒙 𝟐 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Special Trigonometric Limits 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙 =𝟏 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝒙 𝒙 =𝟎 𝐥𝐢𝐦 𝒙→𝟎 𝟏+𝒙 𝟏/𝒙 =𝒆 When expressing 𝒙 in radians and not in degrees Explains the “Squeeze” Theorem 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Why is the Limit of 𝐬𝐢𝐧 𝒙 𝒙 =𝟏 (as 𝒙 approaches to zero) ? 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

Why is the Limit of 𝟏−𝐜𝐨𝐬 𝒙 𝒙 =𝟎, (as 𝒙 approaches to zero)? 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 16 Is there another way of rewriting 𝒕𝒂𝒏⁡(𝒙)? Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙 Split the fraction up so we can isolate and utilize a trigonometric limit 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 16 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙 Utilize the Product Property of Limits 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 17 Try to convert it to one of its trig limits. Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙 Try to get it where the sine trig function to cancel. Whatever is applied to the bottom, must be applied to the top. 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 17 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 18 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Pattern? Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙 =𝟒 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙 = 𝟐 𝟑 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙 = 𝟓 𝟑 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟓𝒙 𝒙 = 𝟓 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟐𝐬𝐢𝐧 𝟑𝒙 𝟓𝒙 = 𝟔 𝟓 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 19 Split the fraction up so we can isolate and utilize a trigonometric limit Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Example 19 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 cos(0) = 1 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟑−𝟑 𝐜𝐨𝐬 𝒙 𝒙 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐬𝐢𝐧 𝒙 𝒙 (A) 𝟎 (B) –𝝅/𝟐 (C) 𝟐 𝟐 /𝝅 (D) 𝟐/𝝅 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐬𝐢𝐧 𝒙 𝒙 Vocabulary Connections and Process Answer and Justifications 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry

§1.3: Properties of Limits with Trigonometry Assignment Page 67 5-21 EOO, 23, 25, 27-35 odd, 37-57 odd (omit 45), 63-73 odd, 89 5/2/2019 11:13 AM §1.3: Properties of Limits with Trigonometry