Evaluating Limits Analytically with Trig Section 1.3A Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/27/2018 5:17 PM §1.3A: 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 Plug in 𝒙 of the function to get the limit 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 What form is this? 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 AS 𝒙 APPROACHES 4, 𝒇(𝒙) OR 𝒚 APPROACHES 8. 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 1 (Calculator) Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 2 Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝟐𝒙 𝟐 −𝒙−𝟑 𝒙 𝟐 −𝟐𝒙−𝟑 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 3 Solve 𝐥𝐢𝐦 𝒙→𝒂 𝒙−𝒂 𝒙 𝟑 − 𝒂 𝟑 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒕→−𝟏 𝒕 𝟑 −𝒕 𝒕 𝟐 −𝟏 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

When in Algebra… NO RADICALS IN THE DENOMINATOR You learned to: NO RADICALS IN THE DENOMINATOR IN LIMITS, NO RADICALS IN THE NUMERATOR and DENOMINATOR 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 What form is this? 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 NO NEED TO FOIL THE BOTTOM 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 5 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟏 −𝟏 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑 . Hint: Don’t combine like terms to the denominator, too early 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 What form is this? 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 7 Evaluate 𝐥𝐢𝐦 𝒙→𝟏 𝟏 𝒙+𝟏 − 𝟏 𝟐 𝒙−𝟏 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: 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. 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry “Squeeze Theorem” 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 8 Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝒄 𝒈(𝒙) where 𝒄=𝟏 for 𝟑𝒙≤𝒈 𝒙 ≤ 𝒙 𝟑 +𝟐 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 8 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 9 Use the Squeeze Theorem to evaluate 𝐥𝐢𝐦 𝒙→𝟒 𝒇(𝒙) for 𝟒𝒙−𝟗≤𝒇 𝒙 ≤ 𝒙 𝟐 −𝟒𝒙+𝟕 for which 𝒙≥𝟎 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: 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 11/27/2018 5:17 PM §1.3A: 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+𝟏/𝒙 𝒙 =𝒆 When expressing 𝒙 in radians and not in degrees 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

Why is the Limit of 𝐬𝐢𝐧 𝒙 𝒙 =𝟏 (as x approaches to zero) ? 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

Why is the Limit of 𝟏−𝐜𝐨𝐬 𝒙 𝒙 =𝟎, (as 𝒙 approaches to zero)? 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 10 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 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 10 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐭𝐚𝐧 𝒙 𝒙 Utilize the Product Property of Limits 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 11 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. 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 11 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟒𝒙 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 12 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝟑𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟓𝐬𝐢𝐧 𝒙 𝟑𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: 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 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 13 Split the fraction up so we can isolate and utilize a trigonometric limit Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Example 13 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏− 𝐜𝐨𝐬 𝟐 𝒙 𝒙 cos(0) = 1 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟑−𝟑 𝐜𝐨𝐬 𝒙 𝒙 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

AP Multiple Choice Practice Question 1 (non-calculator) If 𝒂≠𝟎, then determine 𝐥𝐢𝐦 𝒙→𝒂 𝒙 𝟐 − 𝒂 𝟐 𝒙 𝟒 − 𝒂 𝟒 (A) 𝟏 𝒂 𝟐 (B) 𝟏 𝟐𝒂 𝟐 (C) 𝟏 𝟔𝒂 𝟐 (D) 𝟎 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

AP Multiple Choice Practice Question 1 (non-calculator) If 𝒂≠𝟎, then determine 𝐥𝐢𝐦 𝒙→𝒂 𝒙 𝟐 − 𝒂 𝟐 𝒙 𝟒 − 𝒂 𝟒 Vocabulary Connections and Process Answer and Justifications 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry

§1.3A: Properties of Limits with Trigonometry Assignment Page 67 27-35 odd, 41-57 odd, 63-73 odd, 89 11/27/2018 5:17 PM §1.3A: Properties of Limits with Trigonometry